Evaluate the integral by converting to polar coordinates

Asked by maham237 @ 20/02/2021 in Mathematics viewed by 276 People

Evaluate the iterated integral by converting to polar coordinates. (give an exact answer.) int_(-a)^a int_0^(sqrt(a^2-y^2)) (x^2 + y^2)^(3\/2) text( ) dx text( ) dy

Answered by maham237 @ 20/02/2021

The given integral is
$\int_{-a}^{a} dy \int_{0}^{ \sqrt{a^{2}-y^{2}} } (x^{2}+y^{2})^{3/2} dx$

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
$\int_{- \pi }^{ \pi } d\theta \int_{0}^{a} (r^{2})^{3/2} r \, dr \\\\ = 2 \pi [ \frac{r^{5}}{5}]_{0}^{a} \\\\ = \frac{2 \pi a^{5}}{5}$

Answer: $\frac{2 \pi a^{5}}{5}$

Evaluate the integral by converting to polar coordinates

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