# Calculus/Area

 ← Integration/Exercises Calculus Volume → 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.

 ← Integration/Exercises Calculus Volume → Area