Question

Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y = 10 x 2 and y = x 2 + 2 . Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?

Answer 1

Answer:$A=\frac{14}{9}\cdot \sqrt{2}\ unit^2$

Step-by-step explanation:

Given

Two curves are given

$y=10 x^2$

and $y=x^2+2$

the two curves intersect at

$10x^2=x^2+2$

$x=\pm \frac{\sqrt{2}}{3}$

to get the we need to integrate the curves over x axis

$A=\int_{-\frac{\sqrt{2}}{3}}^{\frac{\sqrt{2}}{3}}\left ( x^2+3-10x^2\right )dx$

$A=2\int_{0}^{\frac{\sqrt{2}}{3}}\left ( 3-9x^2\right )dx$

$A=2\left [ 3\left ( \frac{\sqrt{2}}{3}\right )-\frac{9}{3}\left ( \frac{\sqrt{2}}{3}\right )^3\right ]$

$A=2\sqrt{2}\left [ 1-\frac{2}{9}\right ]$

$A=\frac{14}{9}\cdot \sqrt{2}\ unit^2$