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Substitution is often used to evaluate integrals involving exponential functions or logarithms. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Formulas for derivatives of inverse trigonometric functions developed inDerivatives of Exponential and Logarithmic Functionslead directly to integration formulas involving inverse trigonometric functions. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Substitution is often required to put the integrand in the correct form. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
â i = 1 n c = n c â i = 1 n c = n c â i = 1 n c a i = c â i = 1 n a i â i = 1 n c a i = c â i = 1 n a i â i = 1 n ( a i + b i ) = â i = 1 n a i + â i = 1 n b i â i = 1 n ( a i + b i ) = â i = 1 n a i + â i = 1 n b i â i = 1 n ( a i â b i ) = â i = 1 n a i â â i = 1 n b i â i = 1 n ( ... | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â i = 1 n i = 1 + 2 + ⯠+ n = n ( n + 1 ) 2 â i = 1 n i = 1 + 2 + ⯠+ n = n ( n + 1 ) 2 â i = 1 n i 2 = 1 2 + 2 2 + ⯠+ n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 â i = 1 n i 2 = 1 2 + 2 2 + ⯠+ n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 â i = 0 n i 3 = 1 3 + 2 3 + ⯠+ n 3 = n 2 ( n + 1 ) 2 4 â i = 0 n i 3 = 1 3 + 2 3 ... | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
A â L n = f ( x 0 ) Î x + f ( x 1 ) Î x + ⯠+ f ( x n â 1 ) Î x = â i = 1 n f ( x i â 1 ) Î x A â L n = f ( x 0 ) Î x + f ( x 1 ) Î x + ⯠+ f ( x n â 1 ) Î x = â i = 1 n f ( x i â 1 ) Î x | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
A â R n = f ( x 1 ) Î x + f ( x 2 ) Î x + ⯠+ f ( x n ) Î x = â i = 1 n f ( x i ) Î x A â R n = f ( x 1 ) Î x + f ( x 2 ) Î x + ⯠+ f ( x n ) Î x = â i = 1 n f ( x i ) Î x | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« a b f ( x ) d x = lim n â â â i = 1 n f ( x i * ) Î x â« a b f ( x ) d x = lim n â â â i = 1 n f ( x i * ) Î x | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« a a f ( x ) d x = 0 â« a a f ( x ) d x = 0 â« b a f ( x ) d x = â â« a b f ( x ) d x â« b a f ( x ) d x = â â« a b f ( x ) d x â« a b [ f ( x ) + g ( x ) ] d x = â« a b f ( x ) d x + â« a b g ( x ) d x â« a b [ f ( x ) + g ( x ) ] d x = â« a b f ( x ) d x + â« a b g ( x ) d x â« a b [ f ( x ) â g ... | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
If f ( x ) f ( x ) is continuous over an interval [ a , b ] , [ a , b ] , then there is at least one point c â [ a , b ] c â [ a , b ] such that f ( c ) = 1 b â a â« a b f ( x ) d x . f ( c ) = 1 b â a â« a b f ( x ) d x . | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
If f ( x ) f ( x ) is continuous over an interval [ a , b ] , [ a , b ] , and the function F ( x ) F ( x ) is defined by F ( x ) = â« a x f ( t ) d t , F ( x ) = â« a x f ( t ) d t , then F â² ( x ) = f ( x ) . F â² ( x ) = f ( x ) . | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
If f is continuous over the interval [ a , b ] [ a , b ] and F ( x ) F ( x ) is any antiderivative of f ( x ) , f ( x ) , then â« a b f ( x ) d x = F ( b ) â F ( a ) . â« a b f ( x ) d x = F ( b ) â F ( a ) . | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
F ( b ) = F ( a ) + â« a b F ' ( x ) d x F ( b ) = F ( a ) + â« a b F ' ( x ) d x or â« a b F ' ( x ) d x = F ( b ) â F ( a ) â« a b F ' ( x ) d x = F ( b ) â F ( a ) | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« f [ g ( x ) ] g â² ( x ) d x = â« f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C â« f [ g ( x ) ] g â² ( x ) d x = â« f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« a b f ( g ( x ) ) g ' ( x ) d x = â« g ( a ) g ( b ) f ( u ) d u â« a b f ( g ( x ) ) g ' ( x ) d x = â« g ( a ) g ( b ) f ( u ) d u | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« e x d x = e x + C â« e x d x = e x + C â« a x d x = a x ln a + C â« a x d x = a x ln a + C | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« x â1 d x = ln | x | + C â« x â1 d x = ln | x | + C â« ln x d x = x ln x â x + C = x ( ln x â 1 ) + C â« ln x d x = x ln x â x + C = x ( ln x â 1 ) + C â« log a x d x = x ln a ( ln x â 1 ) + C â« log a x d x = x ln a ( ln x â 1 ) + C | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
â« d u a 2 â u 2 = sin â1 ( u a ) + C â« d u a 2 â u 2 = sin â1 ( u a ) + C â« d u a 2 + u 2 = 1 a tan â1 ( u a ) + C â« d u a 2 + u 2 = 1 a tan â1 ( u a ) + C â« d u u u 2 â a 2 = 1 a sec â1 ( u a ) + C â« d u u u 2 â a 2 = 1 a sec â1 ( u a ) + C | https://openstax.org/books/calculus-volume-2/pages/1-key-equations |
average value of a function : (orfave) the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
change of variables : the substitution of a variable, such asu, for an expression in the integrand | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
definite integral : a primary operation of calculus; the area between the curve and thex-axis over a given interval is a definite integral | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
fundamental theorem of calculus : the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
fundamental theorem of calculus, part 1 : uses a definite integral to define an antiderivative of a function | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
fundamental theorem of calculus, part 2 : (also,evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
integrable function : a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums asngoes to infinity exists | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
integrand : the function to the right of the integration symbol; the integrand includes the function being integrated | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
integration by substitution : a technique for integration that allows integration of functions that are the result of a chain-rule derivative | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
left-endpoint approximation : an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
limits of integration : these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
lower sum : a sum obtained by using the minimum value off(x)f(x)on each subinterval | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
mean value theorem for integrals : guarantees that a pointcexists such thatf(c)f(c)is equal to the average value of the function | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
net change theorem : if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
net signed area : the area between a function and thex-axis such that the area below thex-axis is subtracted from the area above thex-axis; the result is the same as the definite integral of the function | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
partition : a set of points that divides an interval into subintervals | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
regular partition : a partition in which the subintervals all have the same width | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
riemann sum : an estimate of the area under the curve of the formAââi=1nf(xi*)ÎxAââi=1nf(xi*)Îx | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
right-endpoint approximation : the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
sigma notation : (also,summation notation) the Greek letter sigma (Σ) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
total area : total area between a function and thex-axis is calculated by adding the area above thex-axis and the area below thex-axis; the result is the same as the definite integral of the absolute value of the function | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
upper sum : a sum obtained by using the maximum value off(x)f(x)on each subinterval | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
variable of integration : indicates which variable you are integrating with respect to; if it isx, then the function in the integrand is followed bydx | https://openstax.org/books/calculus-volume-2/pages/1-key-terms |
Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
To find the area between two curves defined by functions, integrate the difference of the functions. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Sometimes it can be easier to integrate with respect toyto find the area. The principles are the same regardless of which variable is used as the variable of integration. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
For solids of revolution, the volume slices are often disks and the cross-sections are circles. The method of disks involves applying the method of slicing in the particular case in which the cross-sections are circles, and using the formula for the area of a circle. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
If a solid of revolution has a cavity in the center, the volume slices are washers. With the method of washers, the area of the inner circle is subtracted from the area of the outer circle before integrating. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. This method is sometimes preferable to either the method of disks or the method of washers because we integrate with respect to the other variable. In some cases, one integral is substantia... | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The geometry of the functions and the difficulty of the integration are the main factors in deciding which integration method to use. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The arc length of a curve can be calculated using a definite integral. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions ofy.y. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Several physical applications of the definite integral are common in engineering and physics. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Definite integrals can be used to determine the mass of an object if its density function is known. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
For point masses distributed along a number line, the moment of the system with respect to the origin isM=âi=1nmixi.M=âi=1nmixi.For point masses distributed in a plane, the moments of the system with respect to thex- andy-axes, respectively, areMx=âi=1nmiyiMx=âi=1nmiyiandMy=âi=1nmixi,My=âi=1nmixi,respective... | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
For a lamina bounded above by a functionf(x),f(x),the moments of the system with respect to thex- andy-axes, respectively, areMx=Ïâ«ab[f(x)]22dxMx=Ïâ«ab[f(x)]22dxandMy=Ïâ«abxf(x)dx.My=Ïâ«abxf(x)dx. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Thex- andy-coordinates of the center of mass can be found by dividing the moments around they-axis and around thex-axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The cornerstone of the development is the definition of the natural logarithm in terms of an integral. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The functionexexis then defined as the inverse of the natural logarithm. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
General exponential functions are defined in terms ofex,ex,and the corresponding inverse functions are general logarithms. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Familiar properties of logarithms and exponents still hold in this more rigorous context. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Exponential growth and exponential decay are two of the most common applications of exponential functions. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Systems that exhibit exponential growth follow a model of the formy=y0ekt.y=y0ekt. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
In exponential growth, the rate of growth is proportional to the quantity present. In other words,yâ²=ky.yâ²=ky. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Systems that exhibit exponential growth have a constant doubling time, which is given by(ln2)/k.(ln2)/k. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Systems that exhibit exponential decay follow a model of the formy=y0eâkt.y=y0eâkt. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Systems that exhibit exponential decay have a constant half-life, which is given by(ln2)/k.(ln2)/k. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Hyperbolic functions are defined in terms of exponential functions. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
With appropriate range restrictions, the hyperbolic functions all have inverses. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
The most common physical applications of hyperbolic functions are calculations involving catenaries. | https://openstax.org/books/calculus-volume-2/pages/2-key-concepts |
A = â« a b [ f ( x ) â g ( x ) ] d x A = â« a b [ f ( x ) â g ( x ) ] d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
A = â« c d [ u ( y ) â v ( y ) ] d y A = â« c d [ u ( y ) â v ( y ) ] d y | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
V = â« a b Ï [ f ( x ) ] 2 d x V = â« a b Ï [ f ( x ) ] 2 d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
V = â« c d Ï [ g ( y ) ] 2 d y V = â« c d Ï [ g ( y ) ] 2 d y | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
V = â« a b Ï [ ( f ( x ) ) 2 â ( g ( x ) ) 2 ] d x V = â« a b Ï [ ( f ( x ) ) 2 â ( g ( x ) ) 2 ] d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
V = â« a b ( 2 Ï x f ( x ) ) d x V = â« a b ( 2 Ï x f ( x ) ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
Arc Length = â« a b 1 + [ f â² ( x ) ] 2 d x Arc Length = â« a b 1 + [ f â² ( x ) ] 2 d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
Arc Length = â« c d 1 + [ g â² ( y ) ] 2 d y Arc Length = â« c d 1 + [ g â² ( y ) ] 2 d y | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
Surface Area = â« a b ( 2 Ï f ( x ) 1 + ( f â² ( x ) ) 2 ) d x Surface Area = â« a b ( 2 Ï f ( x ) 1 + ( f â² ( x ) ) 2 ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
m = â« a b Ï ( x ) d x m = â« a b Ï ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
m = â« 0 r 2 Ï x Ï ( x ) d x m = â« 0 r 2 Ï x Ï ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
W = â« a b F ( x ) d x W = â« a b F ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
F = â« a b Ï w ( x ) s ( x ) d x F = â« a b Ï w ( x ) s ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
m = Ï â« a b f ( x ) d x m = Ï â« a b f ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
M x = Ï â« a b [ f ( x ) ] 2 2 d x and M y = Ï â« a b x f ( x ) d x M x = Ï â« a b [ f ( x ) ] 2 2 d x and M y = Ï â« a b x f ( x ) d x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
x â = M y m and y â = M x m x â = M y m and y â = M x m | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
ln x = â« 1 x 1 t d t ln x = â« 1 x 1 t d t Z | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
Exponential function y = e x y = e x | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
ln y = ln ( e x ) = x ln y = ln ( e x ) = x Z | https://openstax.org/books/calculus-volume-2/pages/2-key-equations |
arc length : the arc length of a curve can be thought of as the distance a person would travel along the path of the curve | https://openstax.org/books/calculus-volume-2/pages/2-key-terms |
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