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 have the iterated integral:

$$\int_0^1\int_x^1 e^{y^2}\,dy\,dx$$

To change the order of integration, we integrate with respect to y first, then with respect to x. This gives:

\begin{align*}
\int_0^1\int_x^1 e^{y^2}\,dy\,dx &= \int_0^1\int_0^y e^{y^2}\,dx\,dy \
&= \int_0^1 y e^{y^2}\,dy \
&= \frac{1}{2}\int_0^1 2y e^{y^2}\,dy \
&= \frac{1}{2}\int_1^{e} e^{u}\,du \quad (u=y^2, du=2y\,dy) \
&= \frac{1}{2}(e-1) \
\end{align*}

Therefore, the answer is (C).

Evaluate the Iterated Integral by First Changing the Order of Integration