Finding the volume bounded by a cylinder and a plane

Question

I have been given the following equations:

$$x^2 + z^2 = 9$$ $$ x = 0 $$ $$ y = 0 $$ $$ z = 0 $$ $$ x + 2y = 2 $$

and have been asked to find the volume of the bounded region. I understand the principle of the integration and how to apply it in this scenario, but I tend to have trouble determining the bounds of integration for a three dimensional region such as the one described above. I have tried sketching the cylinder and the plane, but to no avail; I'm consistently unable to determine the bounds. In general, how would one go about determining the integral bounds for a region such as the above?

Answer 1

Apologies in advance, for bad English. You need to start off, by integrating left, integrating right and combining. For example, z component vector, you can integrate first in this case (because it is involved in the least equations). Then, you may go about, triple integrate the other ones (for example, x component first then y component). You will easily be able to determine the integral bounds after this, as you've integrated all three components.

Answer 2

Sketching is indeed a good idea when you deal with non-trivial boundaries. In this case, the main boundaries are the coordinate planes $x=0$, $y=0$, and $z=0$. Then, by inspection, we notice that the given plane $x = 2- 2y$ makes the bound region (in $x$-$y$ plane) in the first octant, which means that our boundaries become

$x\in[0, 2-2y]$, $y\in [0, 1]$ and $z \in [0, \sqrt{9-x^2}]$.

The latter boundary appears due to the fact that the cylinder is oriented along the $z$-axis.

I guess practice of working with this type of problems bring essential experience and then each new problem will look easier and easier.

Hope this helps.