I am trying to solve the problem:
Find the volume below the graph of the function $z=e^{y^2}$ and above $z=0$, while $D:\{(x,y) \ | \ 0\le x \le 1 \ \ \text{and} \ \ x\le y\le 1\}$.
Trying to use triple integral $\mathrm dz \mathrm dy \mathrm dx$, limits for $z$ are $0$ to $e^{y^2}$ and for $y$ limits are $x$ to $1$ and for $x$ is $0$ to $1$.
I am not getting the right answer, it should be $0.5e-0.5$.
Thanks!
Exchange the order of integration: from $0 \le x \le 1$ and $x \le y \le 1$, we obtain $0 \le y \le 1$ and $0 \le x \le y$. So: $$\iint_D e^{y^2}\text{d}x\text{d}y=\int_0^1 \left(\int_0^y e^{y^2}\text{d}x\right)\text{d}y=\int_0^1 ye^{y^2}\text{d}y=\left[\frac{1}{2}e^{y^2}\right]_0^1=\frac{1}{2}(e-1)$$