How can I find the volume of the solid bounded by the cylinder $y^2+z^2=9$ and the planes $x=2y$, $x=0$, $z=0$ in the first octant?
I'm not really sure how I should even approach this, but I know I should somehow solve it using a double integral. I know how to solve double integrals, but I need help setting up a definite double integral for this problem.
Best Answer
If your outer integral is over $z$ (you get to pick-the answer should come out the same) you need to figure out the range of $z$. In this case it runs from $0$ to $3$ so we have $\int_0^3 dz ($something). For the next integral, we get to consider $z$ to be fixed. If we do it over $y$, we need to figure out the range of $y$ at a given $z$. Now we have $\int_0^3 dz \int_{y_{min}}^{y_{max}} dy ($range of $x$ at this $(y,z))$. Now, considering $y$ and $z$ are fixed, you need to figure out the range of $x$, put that in the parentheses and you have your double integral.