Evaluate the iterated integral by first changing the order of integration.

Question

Quizplus · Accepted Answer

The integral is given by 
$$\int_0^2 \int_{x/2}^x e^{y^2} \ dy \ dx.$$
We sketch the region of integration in the $xy$-plane (shaded in orange), which is a triangle bounded by $y=x/2$, $y=x$, and $x=2$.

Now, we change the order of integration. The region of integration is bounded by $0 \leq x \leq 2$ and $y/2 \leq x \leq 2$. Therefore, we have
\begin{align*}
\int_0^2 \int_{x/2}^x e^{y^2} \ dy \ dx &= \int_0^1 \int_0^{2y} e^{y^2} \ dx \ dy + \int_1^2 \int_{y/2}^y e^{y^2} \ dx \ dy \
&= \int_0^1 2ye^{y^2} \ dy + \int_1^2 (y- y/2) e^{y^2} \ dy \
&= e-1 + \frac{1}{2}(e^4 - e) \
&= \boxed{	extbf{(D) } 11eaa948_cd7f_1ab7_84bc_0df0fceaa776_TB2342_11}.
\end{align*}

Evaluate the Iterated Integral by First Changing the Order of Integration