Chapter 8.2, Problem 8E

To determine

To Find: The area of the region enclosed by the graphs of the two functions

Answer to Problem 8E

The area of the region enclosed by the graphs of the two functions $\approx 4.33$

Explanation of Solution

Given information:

Two function

  $y=\cos 2x,y=x^2-2$

Concept used: If $f(x)$ and $g(x)$ are continuous with $f(x)\ge g(x)$ throughout $[a,b]$ , then the area between the curves $y=f(x)$ and $y=g(x)$ from a to b is the integral of $[f(x)-g(x)]$ from a to b

  $A={\displaystyle \underset{a}{\overset{b}{\int }}[f(x)-g(x)]}dx$

Calculation:

Two functions

  $y=\cos 2x,y=x^2-2$

Using a calculator, solve the equation

  $\cos 2x=x^2-2$

  

The solution are $-1.1529611476423$ and $1.1529611461673$

The area is

  $\begin{array}{l}A={\displaystyle \underset{-1.153}{\overset{1.153}{\int }}\left[\cos 2x-x^2+2\right]}dx\\ A={\left[\frac{\sin 2x}{2}-\frac{x^3}{3}+2x\right]}_{-1.153}^{1.153}\\ A=\left[\frac{\sin 2\times 1.153}{2}-\frac{{(1.153)}^3}{3}+2\times 1.153+\frac{\sin 2\times 1.153}{2}-\frac{{(1.153)}^3}{3}+2\times 1.153\right]\\ A\approx 4.33\end{array}$

The area of the region enclosed by the graphs of the two functions $\approx 4.33$