messages listlengths 1 1 | ground_truth stringlengths 3 8 | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 4(-2|(2cos(3π((-2x - 1)/(1x + 1)) + 2) - 2)| + 3)^2 - 3(-2|(2cos(3π((-2x - 1)/(1x + 1)) + 2) - 2)| + 3) - 3 and g(x) = 1|((-2(1e^(-1x)) + 1)/(-2(1e^(-1x)) + 1)) - 3| + 1 between the vertical lines x = -1.2 and x = -1.099999999999999... | 3.7 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 1(-1|((1(1x^2 + 4x + 1) + 2)/(2(1x^2 + 4x + 1) - 1)) - 3|) + 1 and g(x) = 2(5(-1|x + 2| + 2)^2 + 1(-1|x + 2| + 2)) - 4 from x = -3.9 to x = -0.1. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 60.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -2(3|((3(4x - 1) + 1)/(1(4x - 1))) - 3| - 3)^2 - 5(3|((3(4x - 1) + 1)/(1(4x - 1))) - 3| - 3) - 1 and g(x) = (3(5(-4x^2 - 5x + 1) - 5) - 2)/(1(5(-4x^2 - 5x + 1) - 5) + 2) between the vertical lines x = -1.2 and x = -0.3. Round to 1 d... | 1.5 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 2|(-1((1(2x^2 - 5x - 1) + 3)/(-2(2x^2 - 5x - 1) - 3)) + 1) - 3| - 2 and g(x) = -1|(-2cos(1π(-3x - 1) + 3) - 3) + 3| + 3 from x = 0.7 to x = 0.9. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 0.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (3(-1(4x - 2)^2 + 5(4x - 2) - 3))/(2(-1(4x - 2)^2 + 5(4x - 2) - 3) + 1) and g(x) = -5((1x - 2)/(-1x - 3)) - 2 between the vertical lines x = 0.6 and x = 1.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Y... | 4.7 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1e^(-1(1|((1x + 3)/(-2x - 3)) - 3| - 3)) + 2 and g(x) = 2e^(-1(-1x + 5)) - 1 between the vertical lines x = -2.0 and x = -1.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 1.6 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-1(-3|(-3(3x^2 + 2) + 5)| + 1) + 3)/(-2(-3|(-3(3x^2 + 2) + 5)| + 1) - 3) and g(x) = -3(-1cos(2π((-2x - 3)/(3x + 2)) + 3) - 1)^2 + 1(-1cos(2π((-2x - 3)/(3x + 2)) + 3) - 1) + 3 from x = -0.6 to x = -0.5. Round to 1 decimal place.\n\nPresent t... | 0.5 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -4((-2(3sin(1πx - 3)) - 2)/(2(3sin(1πx - 3)) - 1)) - 5 and g(x) = -1(-5x + 3)^2 - 5(-5x + 3) - 5 from x = -0.1 to x = 0.7. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -2e^(1((2(3|x + 1| - 3))/(-2(3|x + 1| - 3) + 3))) + 3, g(x) = -2|(-5x^2 + 2x - 5) + 2| - 1, x = -2.4, and x = 0.3. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 76.9 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 1(-2|((1(2x^2 + 1x - 5) - 1)/(1(2x^2 + 1x - 5)))|) + 5 and g(x) = (-1(2(-3x^2 - 1x - 4) + 3) - 3)/(1(2(-3x^2 - 1x - 4) + 3) + 1) between the vertical lines x = 1.3 and x = 1.4. Round to 1 decimal place.\n\nPresent the answer in LaTe... | 10.0 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -1|(1sin(2π((3(-1x^2 - 3x + 3) - 3)/(-3(-1x^2 - 3x + 3) + 2)) + 2) - 3) + 1| + 2 and g(x) = -1|(-3(2sin(1πx - 2) + 2) + 1) - 3| + 3 from x = -3.6 to x = -3.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 0.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-1(-3|(-1(3x + 4)^2 - 2(3x + 4)) - 2| - 3) + 1)/(1(-3|(-1(3x + 4)^2 - 2(3x + 4)) - 2| - 3) - 3) and g(x) = (-2(2sin(3π(2x + 2) + 1) + 3) + 1)/(1(2sin(3π(2x + 2) + 1) + 3) + 3) between the vertical lines x = 3.3 and x = 3.4. Round t... | 0.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 1|((1(4(-2sin(1πx - 3) + 3) + 3) + 1)/(2(4(-2sin(1πx - 3) + 3) + 3) - 3)) + 1| - 2, g(x) = -2cos(3π(-3|(-5x^2 - 2x - 4) - 2|) - 2) - 1, x = 3.9, and x = 4.0. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxe... | 0.1 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 1((-2(4(-3|x + 2| - 3)^2 - 1(-3|x + 2| - 3) + 5) - 2)/(-3(4(-3|x + 2| - 3)^2 - 1(-3|x + 2| - 3) + 5) + 2)) - 5, g(x) = -5(-5(1|x - 3| - 2) - 4)^2 + 1(-5(1|x - 3| - 2) - 4) + 3, x = 1.6, and x = 2.1. Round your answer to 1 decimal place.\n\nP... | 2.4 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-2(-1|(-1(3cos(1πx - 3))^2 + 4(3cos(1πx - 3)) - 1) + 2| + 3) - 1)/(-3(-1|(-1(3cos(1πx - 3))^2 + 4(3cos(1πx - 3)) - 1) + 2| + 3) + 4) and g(x) = (-3(1(-3|x + 2| + 2)^2 + 1(-3|x + 2| + 2) + 2))/(1(1(-3|x + 2| + 2)^2 + 1(-3|x + 2| + 2) + 2)) f... | 1.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 3sin(1π(5((3(3x^2 + 5x) + 2)/(2(3x^2 + 5x) - 1)) - 5) + 3) + 2, g(x) = -4(3(3|x + 2|)^2 - 4(3|x + 2|)) - 1, x = -1.6, and x = -1.5. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 0.3 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-1(-3(2cos(3πx - 2) + 3)^2 + 2(2cos(3πx - 2) + 3) + 3) + 1)/(2(-3(2cos(3πx - 2) + 3)^2 + 2(2cos(3πx - 2) + 3) + 3) - 1) and g(x) = 3e^(-1(-1x - 4)) - 2 between the vertical lines x = -4.9 and x = -4.2. Round to 1 decimal place.\n\n... | 0.4 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -2sin(2π(1|(3x^2 - 1x - 2) + 3| + 3) - 3), g(x) = 2sin(1π(1x - 3) - 2) - 2, x = 1.7, and x = 1.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 1|(2cos(3π(3((1x - 2)/(1x + 3))^2 - 3((1x - 2)/(1x + 3)) + 4) + 2) - 3) - 3| - 2 and g(x) = -1(3e^(-1((3x + 3)/(1x))) - 1)^2 - 5(3e^(-1((3x + 3)/(1x))) - 1) between the vertical lines x = -3.1 and x = -3.0. Round to 1 decimal place.... | 0.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1|(-4((-1(2x^2 - 3x - 3) - 2)/(2(2x^2 - 3x - 3) + 3)) - 4) - 2| + 3 and g(x) = 2|((2(1e^(2x) - 2) - 3)/(3(1e^(2x) - 2) - 2)) + 2| - 3 between the vertical lines x = 1.9 and x = 2.0. Round to 1 decimal place.\n\nPresent the answer i... | 0.4 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -2((2(3|(-1x^2 + 5x - 3) + 2| - 1) + 1)/(-1(3|(-1x^2 + 5x - 3) + 2| - 1))) - 3 and g(x) = -5(1|(1x^2 - 3x - 5)| + 1) + 3 from x = 0.1 to x = 0.3. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 10.8 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 4(3sin(1π((-1x + 2)/(3x)) + 1) - 1) + 2, g(x) = -2cos(2π(-3x^2 - 3x - 3) + 1) - 3, x = 0.0, and x = 0.1. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.8 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 1(-3((-2(2|x + 1| - 2))/(1(2|x + 1| - 2) - 2))^2 + 5) and g(x) = 2e^(1(-4(3x^2 + 3x - 5) + 2)) from x = -2.4 to x = -1.9. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1.8 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 1cos(1π(-1(-1|x - 2| + 1)^2 - 2(-1|x - 2| + 1) - 1) - 1) - 2 and g(x) = -3(1x + 2)^2 + 3(1x + 2) between the vertical lines x = -2.6 and x = -2.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 0.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (2(1|(5x^2 + 3x) - 2| - 3) + 1)/(2(1|(5x^2 + 3x) - 2| - 3) - 1), g(x) = 5(-2x^2 - 4x + 4) + 3, x = -2.8, and x = -1.4. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 25.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (2(-1cos(1π(-4x^2 - 1x) + 3) + 2) + 2)/(3(-1cos(1π(-4x^2 - 1x) + 3) + 2) - 2) and g(x) = -1cos(2π((2x + 3)/(-2x + 1)) + 3) + 1 between the vertical lines x = -0.0 and x = 0.1. Round to 1 decimal place.\n\nPresent the answer in LaTex... | 0.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (3(-4(-3x + 4)^2 - 5(-3x + 4) - 4) + 2)/(3(-4(-3x + 4)^2 - 5(-3x + 4) - 4)) and g(x) = -2(-1cos(3πx + 3))^2 + 2(-1cos(3πx + 3)) + 1 from x = 0.2 to x = 0.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 0.7 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -5(2(1|((1x)/(2x + 3)) + 3| - 3)^2 - 1) + 3, g(x) = (-1(-2e^(1(-4x^2 + 4x - 4)) - 3) + 1)/(-1(-2e^(1(-4x^2 + 4x - 4)) - 3) - 2), x = -1.4, and x = -0.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{You... | 12.3 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -2cos(2π(1((-3x + 1)/(-2x + 3))^2 - 3((-3x + 1)/(-2x + 3)) + 3) + 1) + 2 and g(x) = 3e^(-1(-4x^2 + 1x)) - 1 between the vertical lines x = -0.0 and x = 0.2. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 0.3 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 2|(-1(1e^(-1(4x^2 - 5x - 2)) - 2) - 3)| - 3 and g(x) = 1(-2((3x)/(1x - 3)) + 4)^2 + 1(-2((3x)/(1x - 3)) + 4) - 4 between the vertical lines x = -2.4 and x = 0.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\box... | 11.5 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 4(3cos(1π((-2x + 2)/(1x - 2))) + 2)^2 + 5(3cos(1π((-2x + 2)/(1x - 2))) + 2) + 5, g(x) = 1(1x + 5)^2 + 5(1x + 5) + 1, x = 2.1, and x = 2.2. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5.4 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (2(1(5x^2 + 5x - 3) - 2) + 3)/(1(1(5x^2 + 5x - 3) - 2) - 3), g(x) = 2(2sin(1πx + 2) - 2), x = -1.7, and x = 0.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11.7 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-3(2|(5(3x - 1)^2 - 3(3x - 1)) + 2| + 3) + 2)/(-2(2|(5(3x - 1)^2 - 3(3x - 1)) + 2| + 3) - 3) and g(x) = (-3(1e^(-1(3x^2 - 1x - 2)) + 1) + 2)/(-2(1e^(-1(3x^2 - 1x - 2)) + 1) - 2) from x = 0.2 to x = 0.6. Round to 1 decimal place.\n\nPresent ... | 0.0 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (2(2(2e^(-1x) - 1)^2 - 5(2e^(-1x) - 1) - 2) - 2)/(2(2(2e^(-1x) - 1)^2 - 5(2e^(-1x) - 1) - 2) + 3) and g(x) = 1|(-1x^2 + 4x - 1) - 1| + 3 from x = -0.6 to x = 0.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 4.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-2(-1e^(-1(5(1|x - 1| + 3) + 4)) + 2))/(-3(-1e^(-1(5(1|x - 1| + 3) + 4)) + 2) - 1) and g(x) = 2|(-2cos(2π(3x) + 3)) + 1| - 1 from x = 4.4 to x = 4.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 0.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 4((-3(1x^2 + 5x + 4))/(3(1x^2 + 5x + 4) + 2)) - 4 and g(x) = -2|((1x - 1)/(1x)) + 1| + 2 between the vertical lines x = -0.4 and x = 0.2. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 86.4 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -3(-1e^(-1(-2|(1x^2 - 4x - 4)|)) + 1) - 1 and g(x) = -1(1sin(1π(3|x + 1| - 1) - 1) + 2)^2 + 2(1sin(1π(3|x + 1| - 1) - 1) + 2) + 2 between the vertical lines x = -0.9 and x = -0.8. Round to 1 decimal place.\n\nPresent the answer in L... | 0.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (-2(-1e^(1(-4x - 2)) + 1) - 1)/(2(-1e^(1(-4x - 2)) + 1) - 1), g(x) = -4((-2x - 3)/(2x + 2)) + 2, x = -1.3, and x = -0.4. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 96.9 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -2cos(2π(-3|(2x^2 - 3x + 3) + 1| + 3) + 1) - 3 and g(x) = -5(1sin(2πx - 1) + 1) + 3 from x = 3.2 to x = 3.3000000000000003. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -4(1cos(2π(3|x + 2| + 3) + 1) + 3)^2 + 1, g(x) = -2|(-1e^(-2x) + 3) - 2| - 2, x = -1.5, and x = -1.3. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3.7 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -2|(2(-1x - 1)^2 - 3(-1x - 1) + 1) - 3| + 3 and g(x) = 1(2|x + 1| - 2) + 3 between the vertical lines x = -0.6 and x = -0.4. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 0.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -4(3((-3x - 3)/(2x - 3)) + 3)^2 - 1(3((-3x - 3)/(2x - 3)) + 3) - 3 and g(x) = -2|((-3x + 1)/(-1x))| + 2 between the vertical lines x = -0.0 and x = 0.1. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 48.5 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1((-2(-1(1|x + 1| - 3)^2 - 2(1|x + 1| - 3) - 1) - 1)/(2(-1(1|x + 1| - 3)^2 - 2(1|x + 1| - 3) - 1))) and g(x) = (-3(-3cos(1π(1x^2 + 5x) + 2) - 2) + 2)/(-2(-3cos(1π(1x^2 + 5x) + 2) - 2) - 3) between the vertical lines x = 0.4 and x =... | 0.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (3(-5(-2cos(1π(3x - 4) - 3) - 2)^2 + 3(-2cos(1π(3x - 4) - 3) - 2) + 2) + 2)/(1(-5(-2cos(1π(3x - 4) - 3) - 2)^2 + 3(-2cos(1π(3x - 4) - 3) - 2) + 2) - 3) and g(x) = 3|(-4(1x + 3)^2 + 2(1x + 3) - 2)| - 3 between the vertical lines x = ... | 0.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 5(3|(3x^2 - 3x - 2) - 1| - 3) + 2, g(x) = 1(-2e^(-1x) + 2) + 5, x = -0.8, and x = -0.4. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2.8 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -4((1(-1|x - 1| + 2) - 1)/(1(-1|x - 1| + 2) - 3)) + 5, g(x) = 2e^(-1(1|x + 1| - 2)) - 1, x = -2.4, and x = -0.1. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9.4 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 4(-1((-2x - 3)/(3x - 2)) + 4)^2 + 4(-1((-2x - 3)/(3x - 2)) + 4) + 5 and g(x) = -2sin(2π(3x^2 + 1x + 1)) + 3 from x = 0.4 to x = 0.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 1.6 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -1((2(3|x - 3| + 3))/(-2(3|x - 3| + 3) - 2))^2 - 2((2(3|x - 3| + 3))/(-2(3|x - 3| + 3) - 2)) - 4, g(x) = -1(3x^2 + 4x - 3) - 3, x = -1.9, and x = 0.5. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 7.0 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 2((-3(3x - 2) - 1)/(1(3x - 2)))^2 + 3((-3(3x - 2) - 1)/(1(3x - 2))) - 4 and g(x) = -1cos(3π((-1x)/(-2x - 2))) + 3 between the vertical lines x = -0.6 and x = -0.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\b... | 0.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (-2(3cos(3π(1|x - 2| - 2) - 1) - 3) + 3)/(2(3cos(3π(1|x - 2| - 2) - 1) - 3)), g(x) = 1(2|x - 1| + 1) - 3, x = 0.7, and x = 1.1. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 0.2 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -2|(1cos(1π(-5(-3x^2 + 5x - 4) - 1) - 2) + 3) + 2| + 3 and g(x) = -5(2|(-2sin(2πx + 2) + 1)| - 3) + 4 from x = 4.5 to x = 4.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.3 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 4(3|((-3x + 1)/(2x - 1))| - 3) + 3, g(x) = 3|(-1x - 1) + 2| - 1, x = -1.9, and x = 0.1. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (1(3|(-3x + 1) + 3|) - 3)/(2(3|(-3x + 1) + 3|) + 2), g(x) = 3(-1|x + 3| + 1)^2 - 3(-1|x + 3| + 1) - 5, x = -4.9, and x = -1.1. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 17.8 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 3(-2|(5x^2 + 5x - 5) + 3| + 2) - 5, g(x) = 1e^(2((1x)/(1x - 2))) - 2, x = -1.3, and x = -1.2. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -4(2cos(2π(2(1|x - 2|) - 2) + 2))^2 - 1, g(x) = -5((-2(-4x^2 - 3x - 5) + 3)/(-1(-4x^2 - 3x - 5) - 2)) + 1, x = -3.0, and x = -2.9. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 0.7 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 2cos(3π(2(-2|(4x - 4) - 3|)^2 - 2(-2|(4x - 4) - 3|) + 1) - 1) - 2, g(x) = -3cos(1π(5(-2|x + 2|)^2 + 5(-2|x + 2|))) + 3, x = 4.3, and x = 4.6. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1.5 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 2|((-3(1(-5x^2 + 4x + 1) - 4) - 1)/(-2(1(-5x^2 + 4x + 1) - 4) + 3)) - 2| - 2 and g(x) = 5(3(-3e^(-1x) + 2)^2 + 3(-3e^(-1x) + 2)) - 5 between the vertical lines x = -0.1 and x = 0.6. Round to 1 decimal place.\n\nPresent the answer in... | 3.9 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -1|(-5(-1((1x + 2)/(2x + 2))^2 + 3((1x + 2)/(2x + 2)) - 1) - 5) - 1| + 2 and g(x) = 5(-1(-2|x| + 3)^2 + 1) + 4 from x = -0.8 to x = -0.7000000000000001. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 0.3 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 3cos(1π(2(-1((-2x - 2)/(-1x + 2)) + 1)^2 - 3(-1((-2x - 2)/(-1x + 2)) + 1)) - 3) + 1 and g(x) = 1|(-1(-3sin(2πx - 2))^2 + 2(-3sin(2πx - 2)) + 2) + 1| - 2 from x = 0.7 to x = 0.8. Round to 1 decimal place.\n\nPresent the answer in LaTex format... | 0.2 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 1cos(3π(-1|((-1(-2x^2 - 1x + 5))/(-1(-2x^2 - 1x + 5) - 1))| + 3) - 1) - 3 and g(x) = 3cos(1π(1|((3x + 2)/(1x + 1))| - 2) + 3) - 1 from x = -1.1 to x = -1.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 0.2 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 2(-3sin(3π((-1x - 3)/(2x + 2)) - 1) + 3)^2 - 4(-3sin(3π((-1x - 3)/(2x + 2)) - 1) + 3) + 5 and g(x) = (1(-1|x + 2| + 1) + 3)/(2(-1|x + 2| + 1) - 1) from x = -2.1 to x = -1.8. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\... | 8.4 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (3(-1(-2|(3x^2 - 3) + 3| + 1) + 1) + 2)/(2(-1(-2|(3x^2 - 3) + 3| + 1) + 1) - 1) and g(x) = 2cos(2π(-3(-1x^2 + 3x) + 2) - 3) + 1 from x = 4.5 to x = 4.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 0.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -2(-3|((-2x - 3)/(-1x + 2))| + 2)^2 - 3(-3|((-2x - 3)/(-1x + 2))| + 2) + 4 and g(x) = (1(-2x - 1))/(1(-2x - 1) + 3) from x = -0.9 to x = -0.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 3.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-2(-1|(-3x + 4) + 2| + 1) + 2)/(1(-1|(-3x + 4) + 2| + 1)) and g(x) = -1sin(2π((2x + 2)/(1x + 3))) + 2 from x = 1.9 to x = 2.1. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 1|(2(-3x^2 + 1x - 1) - 1) + 2| - 3 and g(x) = 2(1cos(3πx + 1) + 3) - 5 from x = -0.6 to x = -0.4. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (1(5(-4(-1|x| - 2)^2 - 4(-1|x| - 2)) + 4))/(1(5(-4(-1|x| - 2)^2 - 4(-1|x| - 2)) + 4)), g(x) = -2((-2(2x + 4) + 2)/(-1(2x + 4) + 3))^2 + 5((-2(2x + 4) + 2)/(-1(2x + 4) + 3)) + 1, x = -1.5, and x = 3.5. Round your answer to 1 decimal place.\n\... | -11.7 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-2(2|(1cos(2πx + 3) - 3)| + 1))/(-2(2|(1cos(2πx + 3) - 3)| + 1) - 2) and g(x) = (2(3|x - 3| - 1) - 1)/(3(3|x - 3| - 1) - 1) between the vertical lines x = 2.7 and x = 3.3. Round to 1 decimal place.\n\nPresent the answer in LaTex fo... | 0.1 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (1(-3(1e^(2x) - 2)^2 - 4(1e^(2x) - 2) + 3) - 2)/(2(-3(1e^(2x) - 2)^2 - 4(1e^(2x) - 2) + 3)), g(x) = (-2(-1x^2 + 4x - 1) + 3)/(1(-1x^2 + 4x - 1)), x = 0.6, and x = 3.3. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex form... | 3.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 5(-2sin(1π((-1x)/(1x - 3)) + 1) + 2)^2 + 5(-2sin(1π((-1x)/(1x - 3)) + 1) + 2) - 4 and g(x) = (-1(-1cos(1πx - 1) - 3) - 3)/(2(-1cos(1πx - 1) - 3) + 1) between the vertical lines x = 4.0 and x = 4.2. Round to 1 decimal place.\n\nPrese... | 0.6 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 1e^(2(5(-2|x| + 2)^2 + 3(-2|x| + 2))) - 2, g(x) = 3e^(-2(-5x + 5)) + 2, x = 0.9, and x = 1.6. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 83.7 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -3|(-3(-5x - 3)^2 - 4(-5x - 3) + 3) - 2|, g(x) = -1|((-2x - 2)/(3x)) + 3| - 1, x = -0.7, and x = -0.6. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.2 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 2sin(2π(-1|((2x - 1)/(1x + 3)) + 1| + 2) - 1) + 2 and g(x) = 3(1sin(3πx + 1) - 1)^2 + 5(1sin(3πx + 1) - 1) + 5 from x = -3.1 to x = -3.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 0.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-2(2|(-4(1x^2 + 5x + 5) - 2)|) - 2)/(1(2|(-4(1x^2 + 5x + 5) - 2)|)) and g(x) = -3|(-2(-1cos(3πx))^2 + 3(-1cos(3πx)) - 2) - 2| - 3 between the vertical lines x = -3.4 and x = -3.3. Round to 1 decimal place.\n\nPresent the answer in ... | 4.8 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 2|(2cos(1π(-4x^2 - 5x + 2) - 2) - 3) - 2| - 1 and g(x) = 2((-1x + 2)/(1x))^2 - 4((-1x + 2)/(1x)) - 2 from x = -3.3 to x = -3.2. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.3 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (3(4(-2cos(1πx - 3) - 1) - 2) - 1)/(-2(4(-2cos(1πx - 3) - 1) - 2) - 3) and g(x) = 2(-2cos(3πx + 2) - 1) - 2 from x = 4.1 to x = 4.2. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 0.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-1(5(-2|x - 1|)^2 - 2) + 3)/(1(5(-2|x - 1|)^2 - 2) - 2) and g(x) = -4(3x - 2)^2 + 3 from x = 0.5 to x = 1.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3.5 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -3(-2sin(2π(1x^2 - 2x + 1) - 2) + 3) - 1, g(x) = -1(-1x - 2)^2 - 4(-1x - 2) + 5, x = -4.6, and x = -4.5. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.4 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 3(-3cos(3π(3|x + 2| + 2) + 2))^2 - 3 and g(x) = -1(-2|x - 3| - 1) + 2 between the vertical lines x = 2.5 and x = 2.6. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.9 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-3(-3(1sin(1πx - 2) + 2)^2 - 1(1sin(1πx - 2) + 2) + 3) - 1)/(-3(-3(1sin(1πx - 2) + 2)^2 - 1(1sin(1πx - 2) + 2) + 3) - 1) and g(x) = -3sin(2π((-2x)/(2x + 1)) + 2) + 3 from x = -0.6 to x = -0.5. Round to 1 decimal place.\n\nPresent the answer... | 0.2 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (1(3|(-2(-1sin(3πx + 3) + 2) + 5)| + 1) + 3)/(-2(3|(-2(-1sin(3πx + 3) + 2) + 5)| + 1) + 3), g(x) = -2(-5((-2x)/(2x - 1)))^2 - 2(-5((-2x)/(2x - 1))) + 4, x = -3.2, and x = -3.0. Round your answer to 1 decimal place.\n\nPresent the answer in L... | 8.9 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -2(1|((2x + 1)/(-2x + 3))| + 2)^2 - 4(1|((2x + 1)/(-2x + 3))| + 2) - 5, g(x) = -2((-2x - 1)/(2x + 2))^2 - 1((-2x - 1)/(2x + 2)) - 1, x = -1.2, and x = -0.9. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed... | -12.9 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 3sin(1π(1(-3|x + 1| + 1) + 5) - 1) + 2 and g(x) = -2|((2x - 3)/(1x - 2)) + 1| + 3 between the vertical lines x = 1.6 and x = 1.8. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 0.4 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1((1(1|x - 3| - 1) + 3)/(-1(1|x - 3| - 1) - 2))^2 + 3((1(1|x - 3| - 1) + 3)/(-1(1|x - 3| - 1) - 2)) and g(x) = 4(2e^(-1x) - 2) - 2 between the vertical lines x = 2.9 and x = 3.1. Round to 1 decimal place.\n\nPresent the answer in L... | 0.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = (1(5(2|x - 3|) + 1) - 1)/(-1(5(2|x - 3|) + 1) - 3), g(x) = -1((3x + 2)/(-2x + 3))^2 + 5((3x + 2)/(-2x + 3)) - 5, x = 0.2, and x = 0.9. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 0.9 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -3(1((2(3|x - 1| + 2) - 1)/(1(3|x - 1| + 2) + 3)) - 3)^2 + 1, g(x) = -1|((-3(-3sin(1πx + 1)) - 2)/(-1(-3sin(1πx + 1)) - 2)) - 1|, x = -0.1, and x = 0.0. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{You... | 25.0 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = 3((1(2|x - 1| + 3) + 3)/(-3(2|x - 1| + 3) - 2)) + 4, g(x) = 1(-2|x - 1| - 1)^2 + 5(-2|x - 1| - 1) + 3, x = -1.0, and x = 3.0. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 17.0 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = 2|((2(-2sin(2π(1x^2 + 1) + 2)) - 1)/(-2(-2sin(2π(1x^2 + 1) + 2)) + 3)) - 2| + 2 and g(x) = 5(-3e^(1(2x + 3)) - 3)^2 - 2(-3e^(1(2x + 3)) - 3) + 3 from x = -3.2 to x = -3.1. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\bo... | 5.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -2|((-1(-4(-1x^2 + 2x + 3) + 4) - 3)/(-1(-4(-1x^2 + 2x + 3) + 4))) + 1| - 2 and g(x) = -5(3|(1x^2 + 5x + 4) - 3| + 1) + 1 between the vertical lines x = -0.7 and x = -0.2. Round to 1 decimal place.\n\nPresent the answer in LaTex for... | 7.6 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -4((1(1cos(1π(3x^2 - 2x - 4) + 1) - 3) - 2)/(3(1cos(1π(3x^2 - 2x - 4) + 1) - 3) - 2)) - 2 and g(x) = 1((-1(1x^2 + 3x + 2) + 1)/(3(1x^2 + 3x + 2) - 2)) + 2 from x = -2.4 to x = -0.6. Round to 1 decimal place.\n\nPresent the answer in LaTex fo... | 9.5 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -1sin(3π(1(1|(-2x - 5) - 2| + 2)^2 + 3(1|(-2x - 5) - 2| + 2) + 2) - 3) - 2, g(x) = 1((-2(-1cos(1πx - 1)) - 1)/(-1(-1cos(1πx - 1)) + 3))^2 + 3((-2(-1cos(1πx - 1)) - 1)/(-1(-1cos(1πx - 1)) + 3)) - 3, x = 4.4, and x = 4.5. Round your answer to ... | 0.1 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-2(-1(-2|x - 1| - 1) + 4) - 3)/(1(-1(-2|x - 1| - 1) + 4) + 3) and g(x) = -1(3sin(1πx - 3) - 1)^2 + 2(3sin(1πx - 3) - 1) + 4 between the vertical lines x = 0.0 and x = 0.9. Round to 1 decimal place.\n\nPresent the answer in LaTex fo... | 9.7 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1(-3sin(2π((2x - 1)/(-3x)) - 3) + 3) and g(x) = -3cos(1π(1x^2 - 5x + 1) + 1) - 2 between the vertical lines x = 1.3 and x = 1.8. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 1.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = 1|(1((-3(3x + 1) - 3)/(1(3x + 1) + 1))^2 + 5((-3(3x + 1) - 3)/(1(3x + 1) + 1)) - 2) + 2| and g(x) = 1((2(1|x - 3| - 2) - 1)/(-3(1|x - 3| - 2) + 2))^2 + 3((2(1|x - 3| - 2) - 1)/(-3(1|x - 3| - 2) + 2)) - 4 between the vertical lines x... | -1.3 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -3|(3((2(-3cos(1πx - 1) - 3))/(2(-3cos(1πx - 1) - 3)))^2 - 3((2(-3cos(1πx - 1) - 3))/(2(-3cos(1πx - 1) - 3))) - 5) + 2| - 3 and g(x) = -3((2(-1x^2 - 2x + 1) + 1)/(2(-1x^2 - 2x + 1) + 2)) + 4 from x = -2.8 to x = 0.8. Round to 1 decimal place... | 54.6 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -3e^(-2(1(2x^2 - 1))) + 1, g(x) = -1(2x^2 + 5x - 5) - 1, x = -3.0, and x = 0.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33.4 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -4(2sin(3π((3(-3|x| - 2) + 1)/(1(-3|x| - 2) - 3))) + 3)^2 + 2(2sin(3π((3(-3|x| - 2) + 1)/(1(-3|x| - 2) - 3))) + 3) + 3 and g(x) = -1e^(2(1(-1x^2 + 2x + 2) - 4)) - 2 between the vertical lines x = -1.0 and x = -0.2. Round to 1 decima... | 33.4 | MATH |
[
{
"content": "Find the area of the region bounded by the curves f(x) = -2(1cos(2π(2|x|)) - 1) - 2, g(x) = 1|(1e^(-1x))| + 1, x = 3.7, and x = 3.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0.1 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = -1(-1((1x)/(-2x - 1)) - 5)^2 + 2 and g(x) = -2|((3x + 2)/(-3x + 1)) - 3| - 1 from x = 0.3 to x = 0.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 87.2 | MATH |
[
{
"content": "Calculate the area enclosed between the functions f(x) = (-2(-1(-1x)^2 + 4(-1x) + 4))/(-1(-1(-1x)^2 + 4(-1x) + 4) + 2) and g(x) = -2(-1sin(3πx + 1) - 3) from x = -4.2 to x = 0.2. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13.2 | MATH |
[
{
"content": "Determine the area of the region enclosed by the graphs of f(x) = -2|((-1(-2x + 5) - 1)/(-1(-2x + 5) + 2)) + 1| + 3 and g(x) = -1e^(1(4x^2 - 4x + 2)) + 1 between the vertical lines x = 1.0 and x = 1.5. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 66.9 | MATH |
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