Please find the area and show the work, thank you.

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=5x2y=5x2 and y=x2+2y=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

I assume that the curves are y = 5x² and y = x²+2. These are symmetric about the y axis, we can take the area between them to be

2 times the Integral from 0 to x(intersect) of (x²+2)-5x² dx.

The point of intersection is when 5x² = x²+2 or x = sqrt(2/4) = 1/sqrt(2)

Final integral is 2x integral from 0 to 1/sqrt(2) of (-4x²+2)dx

2(-(4/3)x³ + 2x) evaluated from 0 to sqrt(2)/2 or

2(-(4/3)(sqrt(2)/4) +sqrt(2)) = 2(2sqrt(2)/3) = (4/3)sqrt(2)

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