Question

Sketch the region enclosed by the given curves.

y = 7 cos(πx), y = 8x2 − 2
Find its area.

Answer 1

area = 14/π +4/3 ≈ 5.78967

Explanation:

You want a sketch and the value of the area enclosed by the curves ...

• y = 7·cos(πx)
• y = 8x² -2

Area

The attached graph shows the curves intersect at x = ±1/2, so those are the limits of integration. The area is symmetrical about the y-axis, so we can just integrate over [0, 1/2] and double the result.

`\displaystyle A=2\int_0^(0.5){(7cos((\pi x))-(8x^2-2))}\,dx=2\left[(7)/(\pi)sin((\pi x))-(8)/(3)x^3+2x\right]_0^(0.5)\\\\\\A=(14)/(\pi)-(2)/(3)+2=\boxed{(14)/(\pi)+(4)/(3)\approx 5.78967}`

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