Evaluate the triple integral $$\iiint_E x\,dV$$ where $E$ is bounded by the paraboloid $x=4y^2+4z^2$ and the plane $x=4$.

I have been analyzing the part of my book where it evaluates triple integrals for paraboloids non stop, but I can't seem to figure out the method for setting it up. (and solving) I have a feeling one of the integrals will be the paraboloid given as an upper bound and the plane given as a lower bound, but I'm not sure how to get the other bounds without having to manually graph a bunch of points till i can see where everything intersects. I remember setting equations to each other to get intersections but I'm not sure how to apply that here. If someone could show me a detailed explanation of how to set this up (and solve) it would help a lot. Thanks.


I have a feeling I'm supposed to put for my outer integral $x$ is from $0$ to $4$, and my inner integrals I use the $\pm$ solutions for $y$ and $z$. Is that right? But I'm not sure how to solve it from here.


Yes, integrating along the axis of the paraboloid is a good idea: $$\int_E x\,dV=\int_{x=0}^4x \left(\int_{E_x}dydz\right) dx$$ where $E_x=\{(y,z):4y^2+4z^2\leq x\}$ which is a disk of radius $\frac{\sqrt{x}}{2}$.

Can you take it from here?

  • $\begingroup$ I'm not sure what you mean with that integral, and I'm not sure I can take it from there. $\endgroup$ – 2316354654 Nov 17 '17 at 14:53
  • $\begingroup$ @2316354654 Does it look better now? $\endgroup$ – Robert Z Nov 17 '17 at 15:00
  • $\begingroup$ i might just be really tired and forgetting some stuff. $\endgroup$ – 2316354654 Nov 17 '17 at 15:02
  • $\begingroup$ @2316354654 The inner integral $\int_{E_x}dydz$ is the area of $E_x$. $\endgroup$ – Robert Z Nov 17 '17 at 15:04
  • $\begingroup$ ok. can you tell me how you got a disk of radius $1/2√x$ $\endgroup$ – 2316354654 Nov 17 '17 at 19:50


Change coordinates from $x\to z\to x$ in this case the shape remains the same: $$z=4y^2+4x^2~~~,~~~z=4$$ Then use substitution cylindrical coordinates.

  • $\begingroup$ this homework problem was from the section before cylindrical coordinates $\endgroup$ – 2316354654 Nov 17 '17 at 14:52
  • $\begingroup$ No matter. You can apply Cartesian coordinates as well . $\endgroup$ – Nosrati Nov 17 '17 at 14:55

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