The given integral is
[tex]\int_{-a}^{a} dy \int_{0}^{ \sqrt{a^{2}-y^{2}} } (x^{2}+y^{2})^{3/2} dx [/tex]
Refer to the figure shown below.
In polar coordinates, the domain of integration is a circle with radius r = [0, a], and θ = [-π, π].
The element of area is
dA = r dr dθ
x² + y² = r²
The integral may be written as
[tex]\int_{- \pi }^{ \pi } d\theta \int_{0}^{a} (r^{2})^{3/2} r \, dr \\\\ = 2 \pi [ \frac{r^{5}}{5}]_{0}^{a} \\\\ = \frac{2 \pi a^{5}}{5} [/tex]
Answer: [tex] \frac{2 \pi a^{5}}{5} [/tex]
