What is the volume of the solid bounded by two paraboloids?

[cbarker1]

Hello Everyone,

I need to find the volume of this solid bounded by the paraboloids z=x^2+y^2 and z=8-x^2-y^2. I need to find the region of integration. I need to find the limits of integration as well. I tried to graph the two surfaces. Thanks

Cbarker1

 

[HallsofIvy]

I presume you saw that the first is a paraboloid with vertex at (0, 0, 0), opening upward, with z-axis as its axis, and every cross section at given z a circle with center at (0, 0, z) and radius $$\sqrt{z}$$. The second is a paraboloid with vertex at (0, 0, 8), opening downward, with z-axis as it axis, and every cross section is a circle with center at (0, 0, z) and radius $$\sqrt{8- z}$$. The two paraboloids will intersect where $$z= x^2+ y^2= 8- x^2+ y^2$$ which reduces to $$x^2+ y^2= 4$$ so the intersection is the circle with center (0, 0, 4), radius 2, in the z= 4 plane.

Since both paraboloids open outward to that circle, you can cover the whole figure by covering that circle. The limits of integration will be
1) x going from -2 to 2 and, for each x, y going from $$-\sqrt{4- x^2}$$ to $$\sqrt{4- x^2}$$
or
2) y going from -2 to 2 and, for each y, x going from $$-\sqrt{4- y^2}$$ to $$\sqrt{4- y^2}$$
or, using polar coordinates,
3) r going form 0 to 2 and $$\theta$$ going from 0 to $$2\pi$$.

 

[cbarker1]

What's the function that I need to integrate?

 

[Prove It]

What's the function that I need to integrate?

To evaluate volume, $\displaystyle \begin{align*} V = \int{\int{\int_R{1\,\mathrm{d}x}\,\mathrm{d}y}\,\mathrm{d}z} \end{align*}$

HallsofIvy has helped you to find the boundaries, and remember that when converting to cylindrical polars, $\displaystyle \begin{align*} \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \to r\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}z \end{align*}$...

 

[cbarker1]

What's about a double integral?

 

[Prove It]

What's about a double integral?

Why not try the triple integral first? If done right, then the right double integral will appear...

 

[cbarker1]

the instruction says to use a double integral to find the volume of the solid.