Find the volume of the solid bounded by the cylinders $y=x^2, y=z^2$ and the plane $y=1$

I think the integral should be: $$\int_0^1\int_{-\sqrt y}^\sqrt y\int_{-\sqrt y}^\sqrt y\ dx\,dz\,dy$$

Could someone tell me if this is correct?

  • 1
    $\begingroup$ Would your integral have been any different for $y=x^4$ and $y=z^4$? $\endgroup$ – Henning Makholm Dec 16 '13 at 4:16
  • $\begingroup$ @HenningMakholm I made a typo on the integral limits. Should have been $-\sqrt y \, to \sqrt y$. Is it correct now? $\endgroup$ – EggHead Dec 16 '13 at 12:46
  • $\begingroup$ @user116056 Corrected my typo on the integral limits. Is it correct now? $\endgroup$ – EggHead Dec 16 '13 at 12:48
  • $\begingroup$ That looks more like it -- though using a double integral to calculate the area of a square with known side length is rather overkill ... $\endgroup$ – Henning Makholm Dec 16 '13 at 12:50

This is not quite right. The region of integration there is not curved but straight like this:

how it looks

Instead, maybe to

use cylindrical coordinates.

Also please look at this question here: Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer).


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