Answer to Question #277190 in Calculus for Air

Question

∫ ∫ 𝑥𝑦𝑑𝑥𝑑𝑦 2𝑦

𝑦

2

Answer 1

In unit circle we need to find the integral value.

$\int \int xydxdy= \int_0^{2\pi}\int_0^1r^3sin\theta cos\theta drd\theta\\ =\int_0^{2\pi}[\frac{r^4}{4}]_0^1sin\theta cos\theta drd\theta\\ =\int_0^{2\pi}\frac{1}{4}sin\theta cos\theta drd\theta\\ =\int_0^{2\pi}\frac{1}{8}2sin\theta cos\theta drd\theta\\ =\int_0^{2\pi}\frac{1}{8}sin(2\theta)drd\theta\\ =\frac{1}{8}[\frac{-cos(2\theta)}{2}]_0^{2\pi}\\ =\frac{1}{16}[-1+1]\\ =0$