# Calculus/Area

## Introduction

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

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

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

${\displaystyle A=\int \limits _{a}^{b}{\bigl (}f(x)-g(x){\bigr )}dx}$

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

## Area between two curves

Suppose we are given two functions ${\displaystyle y_{1}=f(x)}$ and ${\displaystyle y_{2}=g(x)}$ and we want to find the area between them on the interval ${\displaystyle [a,b]}$ . Also assume that ${\displaystyle f(x)\geq g(x)}$ for all ${\displaystyle x}$ on the interval ${\displaystyle [a,b]}$ . Begin by partitioning the interval ${\displaystyle [a,b]}$ into ${\displaystyle n}$ equal subintervals each having a length of ${\displaystyle \Delta x={\frac {b-a}{n}}}$ . Next choose any point in each subinterval, ${\displaystyle x_{i}^{*}}$ . Now we can 'create' rectangles on each interval. At the point ${\displaystyle x_{i}*}$ , the height of each rectangle is ${\displaystyle f(x_{i}^{*})-g(x_{i}^{*})}$ and the width is ${\displaystyle \Delta x}$ . Thus the area of each rectangle is ${\displaystyle {\bigl [}f(x_{i}^{*})-g(x_{i}^{*}){\bigr ]}\Delta x}$ . An approximation of the area, ${\displaystyle A}$ , between the two curves is

${\displaystyle A:=\sum _{i=1}^{n}{\Big [}f(x_{i}^{*})-g(x_{i}^{*}){\Big ]}\Delta x}$ .

Now we take the limit as ${\displaystyle n}$ approaches infinity and get

${\displaystyle A=\lim _{n\to \infty }\sum _{i=1}^{n}{\Big [}f(x_{i}^{*})-g(x_{i}^{*}){\Big ]}\Delta x}$

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

${\displaystyle A=\int \limits _{a}^{b}{\bigl (}f(x)-g(x){\bigr )}dx}$ .

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 ${\displaystyle y_{1}=f(x)}$ and ${\displaystyle y_{2}=g(x)}$ . 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 ${\displaystyle x_{1}=f(y)}$ and ${\displaystyle x_{2}=g(y)}$ on the interval ${\displaystyle [c,d]}$ . Note that ${\displaystyle [c,d]}$ are values of ${\displaystyle y}$ . The derivation of this case is completely identical. Similar to before, we will assume that ${\displaystyle f(y)\geq g(y)}$ for all ${\displaystyle y}$ on ${\displaystyle [c,d]}$ . Now, as before we can divide the interval into ${\displaystyle n}$ subintervals and create rectangles to approximate the area between ${\displaystyle f(y)}$ and ${\displaystyle g(y)}$ . It may be useful to picture each rectangle having their 'width', ${\displaystyle \Delta y}$ , parallel to the y-axis and 'height', ${\displaystyle f(y_{i}^{*})-g(y_{i}^{*})}$ at the point ${\displaystyle y_{i}^{*}}$, parallel to the x-axis. Following from the work above we may reason that an approximation of the area, ${\displaystyle A}$ , between the two curves is

${\displaystyle A:=\sum _{i=1}^{n}{\Big [}f(y_{i}^{*})-g(y_{i}^{*}){\Big ]}\Delta y}$ .

As before, we take the limit as ${\displaystyle n}$ approaches infinity to arrive at

${\displaystyle A=\lim _{n\to \infty }\sum _{i=1}^{n}{\Big [}f(y_{i}^{*})-g(y_{i}^{*}){\Big ]}\Delta y}$ ,

which is nothing more than a definite integral, so

${\displaystyle A=\int \limits _{c}^{d}{\bigl (}f(y)-g(y){\bigr )}dy}$ .

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