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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-1/pages/6-key-concepts |
To find the area between two curves defined by functions, integrate the difference of the functions. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-key-concepts |
The arc length of a curve can be calculated using a definite integral. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-key-concepts |
Several physical applications of the definite integral are common in engineering and physics. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-key-concepts |
The functionexexis then defined as the inverse of the natural logarithm. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-concepts |
Familiar properties of logarithms and exponents still hold in this more rigorous context. | https://openstax.org/books/calculus-volume-1/pages/6-key-concepts |
Exponential growth and exponential decay are two of the most common applications of exponential functions. | https://openstax.org/books/calculus-volume-1/pages/6-key-concepts |
Systems that exhibit exponential growth follow a model of the formy=y0ekt.y=y0ekt. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-key-concepts |
Systems that exhibit exponential decay follow a model of the formy=y0eâkt.y=y0eâkt. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-concepts |
Hyperbolic functions are defined in terms of exponential functions. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-concepts |
With appropriate range restrictions, the hyperbolic functions all have inverses. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-concepts |
The most common physical applications of hyperbolic functions are calculations involving catenaries. | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-key-equations |
m = â« a b Ï ( x ) d x m = â« a b Ï ( x ) d x | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-equations |
W = â« a b F ( x ) d x W = â« a b F ( x ) d x | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-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-1/pages/6-key-equations |
Exponential function y = e x y = e x | https://openstax.org/books/calculus-volume-1/pages/6-key-equations |
ln y = ln ( e x ) = x ln y = ln ( e x ) = x Z | https://openstax.org/books/calculus-volume-1/pages/6-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-1/pages/6-key-terms |
catenary : a curve in the shape of the functiony=acosh(x/a)y=acosh(x/a)is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
center of mass : the point at which the total mass of the system could be concentrated without changing the moment | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
centroid : the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina correspond... | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
cross-section : the intersection of a plane and a solid object | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
density function : a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to... | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
disk method : a special case of the slicing method used with solids of revolution when the slices are disks | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
doubling time : if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by(ln2)/k(ln2)/k | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
exponential decay : systems that exhibit exponential decay follow a model of the formy=y0eâkty=y0eâkt | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
exponential growth : systems that exhibit exponential growth follow a model of the formy=y0ekty=y0ekt | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
frustum : a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
half-life : if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by(ln2)/k(ln2)/k | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
Hookeâs law : this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words,F=kx,F=kx,wherekkis a constant | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
hydrostatic pressure : the pressure exerted by water on a submerged object | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
lamina : a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
method of cylindrical shells : a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
moment : ifnmasses are arranged on a number line, the moment of the system with respect to the origin is given byM=âi=1nmixi;M=âi=1nmixi;if, instead, we consider a region in the plane, bounded above by a functionf(x)f(x)over an interval[a,b],[a,b],then the moments of the region with respect to thex- andy-axes are g... | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
slicing method : a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact valu... | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
solid of revolution : a solid generated by revolving a region in a plane around a line in that plane | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
surface area : the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
symmetry principle : the symmetry principle states that if a regionRis symmetric about a linel, then the centroid ofRlies onl | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
theorem of Pappus for volume : this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis 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-1/pages/6-key-terms |
washer method : a special case of the slicing method used with solids of revolution when the slices are washers | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
work : the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance | https://openstax.org/books/calculus-volume-1/pages/6-key-terms |
The use of sigma (summation) notation of the formâi=1naiâi=1naiis useful for expressing long sums of values in compact form. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
For a continuous function defined over an interval[a,b],[a,b],the process of dividing the interval intonequal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The width of each rectangle isÎx=bâan.Îx=bâan. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Riemann sums are expressions of the formâi=1nf(xi*)Îx,âi=1nf(xi*)Îx,and can be used to estimate the area under the curvey=f(x).y=f(x).Left- and right-endpoint approximations are special kinds of Riemann sums where the values of{xi*}{xi*}are chosen to be the left or right endpoints of the subintervals, respectivel... | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Riemann sums allow for much flexibility in choosing the set of points{xi*}{xi*}at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The definite integral can be used to calculate net signed area, which is the area above thex-axis less the area below thex-axis. Net signed area can be positive, negative, or zero. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The properties of definite integrals can be used to evaluate integrals. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The area under the curve of many functions can be calculated using geometric formulas. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The average value of a function can be calculated using definite integrals. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a valuecsuch thatf(c)f(c)equals the average value of the function. SeeThe Mean Value Theorem for Integrals. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. SeeFundamental Theorem of Calculus, Part 1. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. SeeThe Fundamental Theorem of Calculus, Part 2. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
The area under an even function over a symmetric interval can be calculated by doubling the area over the positivex-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term âsubstitutionâ refers to changing variables or substituting the variableuanddufor appropriate expressions in the integrand. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
When using substitution for a definite integral, we also have to change the limits of integration. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. | https://openstax.org/books/calculus-volume-2/pages/1-key-concepts |
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