Question

Evaluate ∫∫[x^2 + y^2 – a^2 ]dxdy where, x and y varies from –a to a.

(a) –^2⁄3 a^4

(b) –^4⁄3 a^4

(c) –^4⁄3 a^5

(d) –^2⁄3 a^5

I had been asked this question in class test.

Question is taken from Application of Double Integrals topic in division Multiple Integrals of Engineering Mathematics

Answer 1

Correct choice is (b) –^4⁄3 a^4

The explanation is: Equation of circle is given by x^2 + b^2 = a^2

\(\int_{-a}^a \int_{-a}^a [x^2+y^2-a^2 ]dxdy\)

=\(\int_{-a}^a [\frac{x^3}{3}+y^2 x-a^2 x]_a^{-a}dy=\int_{-a}^a [\frac{a^3}{3}+ay^2-a^2 a+\frac{a^3}{3}+y^2 a-a^2 a]dy\)

\(\int_{-a}^a \frac{2a^3}{3}+2ay^2- 2a^3 dy=[\frac{2a^3 y}{3}+\frac{2ay^3}{3}-2a^3 y] _a^{-a}=\frac{4a^4}{3}+\frac{4a^4}{3}-4a^4=-\frac{4}{3} a^4\)