Question

Evaluate the iterated integral:

\(\int\limits_0^4 {\int\limits_0^{\sqrt y } {x{y^2}dxdy} } \)

Short answer

The final answer is 32.

Step-by-step solution

First integrating w.r.t \(x\).

\(\begin{aligned}\int\limits_0^4 {\int\limits_0^{\sqrt y } {x{y^2}dxdy} } &= \int\limits_0^4 {\int\limits_0^{\sqrt y } {{y^2}xdxdy} } \\ &= \int\limits_0^4 {\left( {{y^2}\frac{{{x^2}}}{2}} \right)_0^{\sqrt y }dy} \end{aligned}\)

Now integrating w.r.t \(y\).

\(\begin{aligned}\int\limits_0^4 {\left( {\frac{{{y^3}}}{2} - 0} \right)dy} &= \frac{1}{2}\left( {\frac{{{y^4}}}{4}} \right)_0^4\\ &= \frac{1}{2}\left( {\frac{{256}}{4} - 0} \right)\\ &= 32\end{aligned}\)