Chapter 8.2, Problem 38E

To determine

To Find:

To find the area of the region in the first quadrant bounded by curve $y=\frac{1}{x^2}$ and the lines $y=x,x=2$ and the x-axis.

Answer to Problem 38E

The area of the region in the first quadrant bounded by curve and lines is $1$ .

Explanation of Solution

Given information:

The given function is $y=\frac{1}{x^2}$ , $y=x,x=2$ and the x-axis.

The red curve is the graph of $y=\frac{1}{x^2}$ and the blue line is the graph of $y=x$ and the green line is the graph of $x=2$ .

  

The intersection point is $\left(1,1\right)$ ,

Now take the integral of $x$ from $0$ to $1$ ,then of $x^{-2}$ from $1$ to $2$ ,

Therefore, the area of the region in the first quadrant bounded is,

  $\begin{array}{l}A={\displaystyle \underset{0}{\overset{1}{\int }}xdx+{\displaystyle \underset{1}{\overset{2}{\int }}{x}^{-2}dx}}\\ A={\left[\frac{x^2}{2}\right]}_0^1+{\left[-\frac{1}{x}\right]}_1^2\\ A=\left[\frac{1}{2}-\frac{1}{2}+1\right]\\ A=1\\ \end{array}$

Therefore, the area of the region in the first quadrant bounded by curve and lines is $1$ .