Q:

My book doesn't have the answer. Can anyone help me with volumes of solids by slicing?

How do I find the volume in the first octant bounded by the surfaces $x^2=y+2z$ and $x=2$ use the slice in the figure to compute the volume? 

A:

While you can do it with triple integrals, the slice given hinted at a much simpler answer:
Note that $x^2=y+2z$ has only linear terms in $y$ and $z$, so the constant-$x$ slice in the picture is a triangle with base along $z$-direction of length $x^2/2$ and height along $y$-direction of length $x^2$.  So integrating the slices,
$$
\operatorname{Volume}=\int_0^2 \frac12\cdot\frac12x^2\cdot x^2\,\mathrm{d}x
$$
which you should know how to do.

