How can I solve this Iterated integral

Question

domain E

I've this integral on the domain E: x[0,1], y=x^2

$\int_E x^3sin(xy)\, dxdy$

I dont know how can I solve this through the two different ways:first integration in dx and then in dy, and viceversa.

Thanks a lot

Answer 1

Your domain is: E={ $(x,y) | x\in [0,1], \ 0<y<x^2 $}

Your integral is easier to integrate first in y:

$\begin{align} I &=\int_0^1 \int_0^{x^2} x^3 \sin(xy)\ \mathrm{dy\ dx} \\ &=\int_0^1 [-x^2 \cos(xy)]_0^{x^2}\ \mathrm{ dx} \\ &=\int_0^1 -x^2 \cos(x^3)+x^2\ \mathrm{ dx} \\ &=[-\frac 13 \sin(x^3)]_0^1+[\frac{x^3}3]_0^1\ \mathrm{ dx} \\ &=\frac{1-\sin(1)}3\\ \end{align} $