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=6x2 and y=x2+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:$\frac{8}{3}\times \sqrt{\frac{2}{5}}$

Step-by-step explanation:

Given two upward facing parabolas  with equations

$y=6x^2 & y=x^2+2$

The two intersect at

$6x^2=x^2+2$

$5x^2=2$

$x^2$=$\frac{2}{5}$

x=$\pm \sqrt{\frac{2}{5}}$

area  enclosed by them is given by

A=$\int_{-\sqrt{\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left [ \left ( x^2+2\right )-\left ( 6x^2\right ) \right ]dx$

A=$\int_{\sqrt{-\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left ( 2-5x^2\right )dx$

A=$4\left [ \sqrt{\frac{2}{5}} \right ]-\frac{5}{3}\left [ \left ( \frac{2}{5}\right )^\frac{3}{2}-\left ( -\frac{2}{5}\right )^\frac{3}{2} \right ]$

A=$\frac{8}{3}\times \sqrt{\frac{2}{5}}$