Chapter 7.2, Problem 38E

To determine

To find: The area of the region in the first quadrant bounded by the given curve and line.

Answer to Problem 38E

The area is: $A=1$

Explanation of Solution

Given information:

The line $y=x$

And,

  $x=2$

And curve is $y=1/x^2$

Calculation:

The area that has to be evaluated is shaded in blue. As seen in the image, the area that has to be evaluated can be found by deducting the purple region's area from the triangle's area defined by the lines:

  $\begin{array}{c}y=x,\\ x=2\\ y=0\end{array}$

Firstly need to find the limits of integration. Therefore, by equalize:

  $\begin{array}{c}y=\frac{1}{x^2}\\ y=x\end{array}$

i.e.

  $1=x^3$

Therefore, obtain that only intersection is point $(1,1)$ .Since it is clear that $1/x^2<x$

over the interval $[1,2]$

to find the area of the purple region need to evaluate:

  

Finally, from the fact that the area of the triangle is equal toobtain that the area of the blue region is equal to $2-1=1$ .

The graph is shown below:

  

Therefore, the required area of the region in the first quadrant bounded by the given curve and line is