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

• MHB
• cbarker1
In summary: Why not try that first?In summary, the boundaries of integration are the circles with centers at (0, 0, z) and radii \sqrt{z}. The limits of integration are when x goes from -2 to 2 and y goes from -\sqrt{4- x^2} to \sqrt{4- x^2}. The function to be integrated is the function that covers the circle with center (0, 0, 4) and radius 2.
cbarker1
Gold Member
MHB
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

Last edited:
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 $$2\pi$$.

What's the function that I need to integrate?

Cbarker1 said:
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 said:

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

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

1. What is the formula for finding the volume of a solid?

The formula for finding the volume of a solid is V = lwh, where l is the length, w is the width, and h is the height of the solid.

2. How do you determine the units for volume?

The units for volume are determined by the units used for length, width, and height. For example, if the length, width, and height are measured in meters, then the volume will be in cubic meters (m³).

3. How do you find the volume of an irregularly shaped solid?

To find the volume of an irregularly shaped solid, you can use the displacement method. This involves submerging the solid in a liquid and measuring the amount of liquid displaced. The volume of the solid will be equal to the amount of liquid displaced.

4. Can the volume of a solid change?

Yes, the volume of a solid can change. It can change if the solid is heated or cooled, if it undergoes a chemical reaction, or if it is compressed or expanded.

5. How is finding the volume of a solid useful?

Finding the volume of a solid is useful in many scientific and practical applications. It can be used to determine the amount of material needed for construction or manufacturing, to calculate the displacement of objects in fluids, and to measure the capacity of containers or tanks.

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