# 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

$A=A_{\rm {top}}-A_{\rm {bottom}}$  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 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.

 ← Integration/Exercises Calculus Volume → Area