How to do this integral question?

Evaluate ∫_(−2)^2(x+7)sqrt(4−x^2)dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.

∫_(−2)^2(x+7)sqrt(4−x^2)dx=?

2 Answers
Dec 2, 2017

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Dec 2, 2017

14pi

Explanation:

int_-2^2(x+7)sqrt(4-x^2) dx = int_-2^2x sqrt(4-x^2) dx + 7int_-2^2 sqrt(4-x^2) dx

Now think of the circle centered at the origin with radius 2

y^2+x^2= 2^2

int_-2^2 sqrt(4-x^2) dx represents the upper semicircle area or 1/2pi 2^2 and

int_-2^2 x sqrt(4-x^2) dx = 0 because x sqrt(4-x^2) is an odd function in x in [-2,2]

and the result is

int_-2^2(x+7)sqrt(4-x^2) dx = 7 xx 2pi