How to do this integral question?

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=?#

Dec 2, 2017

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Answer 1 · 2017-12-02T09:44:50

#14pi#

#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#