Applications of Integrals

Area between two curves

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 $f(x)\geq g(x)$ 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 $f(y)\geq g(y)$ 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.

Resources

• Area, Wikibooks: Calculus
• Applications of Integrals - Cal I, Paul's Online Notes
• Applications of Integrals - Cal II, Paul's Online Notes