Circle?

Calculus Level 3

$\int_{C} \int_{C}x^{3}y^{2}\, dx\; dy=\, ?$

where $C$ is a region, $x^2+y^2\leq r^2$ , where $r\in \mathbb R$ .

Notation : $\mathbb R$ denotes the set of real numbers .

The answer is 0.

1 solution

Aditya Narayan Sharma
Jul 31, 2016

Substitute $\displaystyle x=a\cos\theta ,y=a\sin\theta$ , so that $\displaystyle |a|\le r \implies a\in[-r,r]$

The integral is : $\displaystyle I=\int_{-r}^{r}\int_{0}^{2\pi} a^5 \sin^2 \theta \cos^3 \theta d\theta da = \int_{-r}^{r}a^5 da \int_{0}^{2\pi} a^5 \sin^2 \theta \cos^3 \theta d\theta =0$ , Since { $\displaystyle \int_{-r}^{r}a^5 da =0$ }

Good day to you Sir/Ma'am. Apparently no one has commented on your solution, so maybe I'm just the bold one. I'm having trouble justifying your solution. I solved this one by converting to polar coordinates, and I got an even power of the radius inside the integrand. Is there something I'm missing here? I seem to have forgotten how to do a change of variables in a multiple integral.

James Wilson - 3 years, 8 months ago

I think you are correct, James. I also got an even power of r (6). I think the solution posted here just left out the factor of r that accompanies the differentials in the integral when you change from cartesian to polar coordinates, but it doesn't effect the final result since the zero ultimately results from integrating the angular part over the boundary.

Tristan Goodman - 1 year, 4 months ago

Circle?

The answer is 0.

1 solution

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