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A little bit rusty on the topic of surface integrals and perhaps some basic calculus after a while. A little help would be appreciated

Find the volume of the solid under the surface $$z=3x^{2}+y^{2} $$ and above the region bounded by $$y=x$$ and $$y^{2}-y=x.$$

The main problem I'm facing is with the parabola. I've found the parabola to cut the y-intercept at $y=0$ and $y=1$. However, this does not square with the solutions I am looking at.

Any help is appreciated.

Rio Alvarado
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Mathematicing
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  • It is helpful to draw pictures for these types of problems. You need to visualize how these surfaces intersect one another in order to find the bounds of your integrals. – Alex G. Aug 05 '15 at 04:43
  • @AlexG. I did but not exactly sure what am I missing. Clearly, the parabola intercept the y-axis at y=0 and y=1. The equation y=x is just the straight line. I want the area bounded within this region. But clearly, when I attempt to determine the point of intersection, I arrived at y=0 which is a contradiction so something is wrong somewhere. – Mathematicing Aug 05 '15 at 04:48
  • I think you are missing that these are surfaces in three dimensional space, not just lines and parabolas. $y=x$ is a plane, and $y^2-y=x$ is a "parabola-shaped cylinder" or whatever you would call it. You need to see not only how these two intersect, but how they intersect with $z=3x^2+y^2$ as well. – Alex G. Aug 05 '15 at 04:52
  • @AlexG. I don't believe that we need to consider the "sides" of the plane $x=y$ and parabolic cylinder $x=y^2-y$. The volume under the surface $z=3x^2+y^2$. We need not see how they intersect with the surface of interest. – Mark Viola Aug 05 '15 at 04:59

1 Answers1

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It might be easier here to view the region in the $x-y$ plane, which comprises the domain for the integral, having $y$ playing the role of the independent variable and having $x$ as a function of $y$.

The region in the $x-y$ plane is bounded from $x=y^2-y$ to $x=y$ as $y$ goes from $0$ to $2$.

Therefore, the volume integral is

$$V=\int_0^2\int_{y^2-y}^y(3x^2+y^2)dx\,dy$$

Can you finish?

Mark Viola
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