Evaluate the iterated integral by first changing the order of integration.
D) The integral can be evaluated only numerically.

Question

Evaluate the iterated integral by first changing the order of integration. 
D) The integral can be evaluated only numerically.

Quizplus · Accepted Answer

We can evaluate the iterated integral by changing the order of integration as follows:
$$\int_{0}^{1}\int_{x^2}^{x}\sin(y^3)\,dy\,dx=\int_{0}^{1}\int_{\sqrt{y}}^{1}\sin(y^3)\,dx\,dy=\int_{0}^{1}\left[(1-\sqrt{y})\sin(y^3)\right]\,dy$$
This integral can be evaluated analytically, giving us the final answer of $\frac{1}{3\sqrt{2}}(\cos(1)-1+\sin(1))$. Therefore, choice C is correct.

Evaluate the Iterated Integral by First Changing the Order of Integration