Calculating Volume of Solid Bounded by Cylinders and Plane

• VinnyCee
In summary, the problem is to find the volume of a solid bounded by three cylinders, and the correct integral is \int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;dz\;dy\;dx\;=\;12\pi. The next problem involves finding the average value of the function f(x, y, z) = sqrt(xyz) within the same solid, and the correct integral is \frac{1}{12\pi}\;\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 -
VinnyCee
Here is the problem:

Find the volume of the solid that is bounded above by the cylinder $$z = 4 - x^2$$, on the sides by the cylinder $$x^2 + y^2 = 4$$, and below by the xy-plane.

Here is what I have:

$$\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;dz\;dy\;dx\;=\;12\pi$$

Is that correct? I didn't post the many steps for integration, but the integral calulation is correct, I just need to know if I set up the integral right. Thanks again

Yes, that's correct. Of course, it would be easier to do the integration in cylindrial coordinates.

What if f(x, y, z) = sqrt(xyz), how to find average value?

Thanks for the double checking! The next problem uses this same integral and assumes that $$f\left(x, y, z\right) = \sqrt{x\;y\;z}$$. Then it says to setup the integral to find the average value of the function within that solid.

Here is what I have:

$$\frac{1}{12\pi}\;\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;\sqrt{x\;y\;z}\;dz\;dy\;dx$$

Does that look right?

1. What is the formula for calculating the volume of a solid bounded by cylinders and planes?

The formula for calculating the volume of a solid bounded by cylinders and planes is V = πr²h, where r is the radius of the cylinder and h is the height of the cylinder.

2. How do you determine the radius and height of a cylinder in this type of calculation?

The radius and height of the cylinder can be determined by looking at the given dimensions of the solid and identifying the dimensions that correspond to the base and height of the cylinder. If necessary, you can use the Pythagorean theorem to calculate the height of the cylinder.

3. Can the same formula be used for all types of cylinders and planes?

Yes, the formula V = πr²h can be used for calculating the volume of any solid bounded by cylinders and planes, regardless of their size or orientation.

4. How do you account for multiple cylinders and planes in the calculation?

If there are multiple cylinders and planes involved in the solid, you can calculate the volume of each individual cylinder and plane using the formula V = πr²h and then add them together to get the total volume of the solid.

5. Are there any real-world applications for calculating the volume of a solid bounded by cylinders and planes?

Yes, calculating the volume of a solid bounded by cylinders and planes is often used in engineering and construction for determining the volume of various structures, such as pipes, tanks, and buildings.

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