< Calculus


Finding the area between two curves, usually given by two explicit functions, is often useful in calculus.

In general the rule for finding the area between two curves is

A=A_{\rm top}-A_{\rm bottom} or

If f(x) is the upper function and g(x) is the lower function

A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx

This is true whether the functions are in the first quadrant or not.

Area between two curvesEdit

Suppose we are given two functions y_1=f(x) and y_2=g(x) and we want to find the area between them on the interval [a,b] . Also assume that f(x)\ge g(x) for all x on the interval [a,b] . Begin by partitioning the interval [a,b] into n equal subintervals each having a length of \Delta x=\frac{b-a}{n} . Next choose any point in each subinterval, x_i^* . Now we can 'create' rectangles on each interval. At the point x_i* , the height of each rectangle is f(x_i^*)-g(x_i^*) and the width is \Delta x . Thus the area of each rectangle is \bigl[f(x_i^*)-g(x_i^*)\bigr]\Delta x . An approximation of the area, A , between the two curves is

A:=\sum_{i=1}^n \Big[f(x_i^*)-g(x_i^*)\Big]\Delta x .

Now we take the limit as n approaches infinity and get

A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(x_i^*)-g(x_i^*)\Big]\Delta x

which gives the exact area. Recalling the definition of the definite integral we notice that

A=\int\limits_a^b \bigl(f(x)-g(x)\bigr)dx .

This formula of finding the area between two curves is sometimes known as applying integration with respect to the x-axis since the rectangles used to approximate the area have their bases lying parallel to the x-axis. It will be most useful when the two functions are of the form y_1=f(x) and y_2=g(x) . Sometimes however, one may find it simpler to integrate with respect to the y-axis. This occurs when integrating with respect to the x-axis would result in more than one integral to be evaluated. These functions take the form x_1=f(y) and x_2=g(y) on the interval [c,d] . Note that [c,d] are values of y . The derivation of this case is completely identical. Similar to before, we will assume that f(y)\ge g(y) for all y on [c,d] . Now, as before we can divide the interval into n subintervals and create rectangles to approximate the area between f(y) and g(y) . It may be useful to picture each rectangle having their 'width', \Delta y , parallel to the y-axis and 'height', f(y_i^*)-g(y_i^*) at the point y_i^*, parallel to the x-axis. Following from the work above we may reason that an approximation of the area, A , between the two curves is

A:=\sum_{i=1}^n \Big[f(y_i^*)-g(y_i^*)\Big]\Delta y .

As before, we take the limit as n approaches infinity to arrive at

A=\lim_{n\to\infty}\sum_{i=1}^n \Big[f(y_i^*)-g(y_i^*)\Big]\Delta y ,

which is nothing more than a definite integral, so

A=\int\limits_c^d \bigl(f(y)-g(y)\bigr)dy .

Regardless of the form of the functions, we basically use the same formula.

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