This is for my homework assignment in WebAssign. The question states:
Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinder $y = 16x^2$, and the planes $z = 3y$ and $z = 2 + y$.
I've graphed the parabolic cylinder by hand. Since it does not mention any "z", the relationship must hold true for all z. That is, the graph of $y = 16x^2$ will run along the length of the z-axis. The other two functions are planes. I've drawn those as well.
But as you can probably tell, it's quite difficult to imagine the solid trapped by these three functions just using rough sketches of the graphs. It's even more difficult to determine what the bounds of integration should be. Note that this homework corresponds to the section on Double Integrals.
Could someone please point me in the right direction? I'm not looking for a homework solution or any overly-explicit answers. Once I figure out how to set up the problem and find the bounds, I'll be able to take care of the rest on my own.
Thank you!
Here is one way of doing it: ask yourself, what is the projection of the solid in the $yz$ plane ? It is the region bounded by $z=3y$, $z=2+y$, and $y=0$: $$ D= \{(y,z)\;|\; 0 \le y \le 1, 3y \le z \le 2+y \} $$ Now, how does the third variable ($x$) evolve on this region? The answer lies in the equation $y=x^2\; \Leftrightarrow\; x=\pm \sqrt{y}$. Therefore, you can describe the solid as $$ E = \{(x,y,z)\; |\; (y,z) \in D, -\sqrt{y} \le x \le \sqrt{y} \} $$
Can you take it from there? You should find $\frac{16}{15}$ in the end.
Note that you can also solve this problem by projecting the solid in the $xy$ plane. In this case, you get $$ E =\{(x,y,z)\;|\; (x,y)\in D, 3y \le z \le 2+y\} $$ where this time $$ D = \{(x,y)\;|\; -1 \le x \le 1, x^2 \le y \le 1 \} $$
It is an excellent exercise to do the exercise both ways and make sure the results match.