Volume of parabolic cylinder bound by plane

$\begingroup$

Make a sketch of the solid in the first octant bounded by the plane

$x + y = 1$

and the parabolic cylinder

$x^{2} + z = 1$

Calculate the volume of the solid..

I have no idea where to even start :(

$\endgroup$ 2

2 Answers

$\begingroup$

As can be seen from the plot below, we have a flat triangle with $(0,0),(0,1),(1,0)$ on $xy-$plane. So the required limits here is as $$x|_0^1,y|_0^{1-x},z|_0^{1-x^2}$$

$\endgroup$ 4 $\begingroup$

You want to take a triple integral

$$\iiint 1 \, dx \, dy \, dz$$

where $X$ is your solid. This makes sense to compute a volume, since you're adding up one unit to your integral per unit value. So all you need to do is work out what the bounds on those integrals should be so that they exactly cover your solid. (You may want to change the order of the $dx$, $dy$, and $dz$ around if it helps).

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy