2.1 Areas between curves

In Introduction to Integration , we developed the concept of the definite integral to calculate the area below a curve on a given interval. In this section, we expand that idea to calculate the area of more complex regions. We start by finding the area between two curves that are functions of $x,$ beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of $y.$

Area of a region between two curves

Let $f(x)$ and $g(x)$ be continuous functions over an interval $\left[a,b\right]$ such that $f(x)\ge g(x)$ on $\left[a,b\right].$ We want to find the area between the graphs of the functions, as shown in the following figure.

As we did before, we are going to partition the interval on the $x\text{-axis}$ and approximate the area between the graphs of the functions with rectangles. So, for $i=0,1,2\text{,…},n,$ let $P=\left\{x_i\right\}$ be a regular partition of $\left[a,b\right].$ Then, for $i=1,2\text{,…},n,$ choose a point $x_i^{*}\in \left[x_{i-1},x_i\right],$ and on each interval $\left[x_{i-1},x_i\right]$ construct a rectangle that extends vertically from $g(x_i^{*})$ to $f(x_i^{*}).$ [link] (a) shows the rectangles when $x_i^{*}$ is selected to be the left endpoint of the interval and $n=10.$ [link] (b) shows a representative rectangle in detail.

The height of each individual rectangle is $f(x_i^{*})-g(x_i^{*})$ and the width of each rectangle is $\text{Δ}x.$ Adding the areas of all the rectangles, we see that the area between the curves is approximated by

This is a Riemann sum, so we take the limit as $n\to \infty$ and we get

These findings are summarized in the following theorem.

We apply this theorem in the following example.

Finding the area of a region between two curves 1

If R is the region bounded above by the graph of the function $f(x)=x+4$ and below by the graph of the function $g(x)=3-\frac{x}{2}$ over the interval $\left[1,4\right],$ find the area of region $R.$

The region is depicted in the following figure.

We have

$\begin{array}{cc}\hfill A& ={{\displaystyle \int }}_a^b\left[f(x)-g(x)\right]dx\hfill \\ & ={\displaystyle {\int }_1^4\left[\left(x+4\right)-\left(3-\frac{x}{2}\right)\right]dx}={\displaystyle {\int }_1^4\left[\frac{3x}{2}+1\right]dx}\hfill \\ & ={\left[\frac{3x^2}{4}+x\right]|}_1^4=\left(16-\frac{7}{4}\right)=\frac{57}{4}.\hfill \end{array}$

The area of the region is $\frac{57}{4}{\text{units}}^2.$

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