Question:
Calculate the volume of the region in $y > 0$ enclosed by the planes $y = 0, z = 0, z = d − x + y$ and the parabolic cylinder $y=d-{x^2}/d$, where $d$ is a positive valued constant.
I know I'll be using a double integral to find the volume but I'm having a hard time visualising it and I'm not sure what to use for the limits.
I think I integrate with respect to $y$ first, but I'm not sure what the limits would be? would they be $y=0$ and $y=d-{x^2}/d$ ?
$$ \displaystyle \int \left[\int_{y=0}^{y=d-{x^2}/d} (d-x+y) \,dy \right] ~ dx\, ?$$
Not sure what my $x$ limits would be I was thinking either $-d,d$ or maybe $\sqrt{d},d$
Can anyone confirm whether I'm integrating the right thing and help me with my limits? thank you
Best Answer
Your limits for $y$ is correct.
Parabolic cylinder $y = d - \frac{x^2}{d}$ has vertex at $(0, d, 0)$ if we see the projection in xy-plane and intersect x-axis at $ (\pm d, 0, 0)$.
Please note that the plane $z = d - x + y ~$ also intersects x-axis at $(d, 0, 0)$ and for the entire area in the xy-plane bound by the parabolic cylinder and $y = 0$, $z$ is bound below by $z = 0$ and above by the plane $z = d - x + y$.
So the integral to find volume should be,
$ \displaystyle \int_{-d}^d \left[\int_0^{d-{x^2}/d} (d -x + y) ~ dy \right] ~ dx$