Evaluate the integral.
-$$\int _ { 1 } ^ { 3 } \int _ { 0 } ^ { y } x ^ { 2 } y ^ { 2 } d x d y$$
A) $\frac { 364 } { 3 }$
B) $\frac { 350 } { 9 }$
C) $\frac { 350 } { 3 }$
D) $\frac { 364 } { 9 }$

Question

Quizplus · Accepted Answer

We can solve this integral by reversing the order of integration. First let's find the limits for $x$: $0\leq x\leq3$ and $x\leq y\leq3$. Thus,
\begin{align*}
\int_{1}^{3}\int_{0}^{y}x^{2}y^{2}dxdy &= \int_{0}^{3}\int_{x}^{3}x^{2}y^{2}dydx \
&= \int_{0}^{3}\left(\frac{1}{3}x^2y^3ight)\Biggrvert_{x}^{3}dx\
&= \int_{0}^{3}\frac{1}{3}(27x^2-x^{5})\,dx\
&= \frac{1}{3}\left(9x^3-\frac{1}{6}x^6ight)\Biggrvert_{0}^{3}\
&= \frac{364}{9}
\end{align*}Therefore, the answer is $\boxed{	extbf{(D)}}$.

Evaluate the Integral ∫13∫0yx2y2dxdy\int _ { 1 } ^ { 3 } \int _ { 0 } ^ { y } x ^ { 2 } y ^ { 2 } d x d y∫13​∫0y​x2y2dxdy

Evaluate the Integral $\int _ { 1 } ^ { 3 } \int _ { 0 } ^ { y } x ^ { 2 } y ^ { 2 } d x d y$