Calculating Average Value of f(x,y,z) in Solid Bounded by Cylinders

• VinnyCee
In summary, the conversation discusses finding the volume of a solid bounded by two cylinders and the xy-plane, and then using that integral to find the average value of a given function within the solid. The person also mentions having a final exam and a research paper and presentation on superconductors.
VinnyCee
Here is the problem:

First Part (already done): 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.

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

Using the integral worked out above, and assuming that $$f\left(x, y, z\right) = \sqrt{x\;y\;z}$$. 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?

Assuming your first integral is correct, which by initial inspection, I believe it is, then yes, your second solution is indeed correct.

BTW so many of your posts have been about multivar. calc! I can tell you might have a pretty big exam coming up!

:D

Wait wait nm, I meant

D:

Big exam coming

Indeed, I do have a final coming up in about a week! However, I only have one more problem to check here and then I will be concentrating on my Research Paper and Presentation for the rest of this week and upcoming weekend. It is supposed to be about superconductors and their future applications in computing. I have to present on Monday.

What is the formula for calculating the average value of f(x,y,z) in a solid bounded by cylinders?

The formula for calculating the average value of f(x,y,z) in a solid bounded by cylinders is:
Average value = (1/V)∫∫∫ f(x,y,z) dV, where V is the volume of the solid.

How do I determine the limits of integration for calculating the average value?

The limits of integration can be determined by finding the intersection points between the cylinders that bound the solid. These points will serve as the boundaries for the triple integrals.

Can the average value of f(x,y,z) in a solid bounded by cylinders be negative?

Yes, the average value of f(x,y,z) can be negative. This can occur if the function has negative values within the solid. The average value represents the overall "average" of the function within the solid, regardless of whether the values are positive or negative.

What are some real-life applications of calculating the average value of f(x,y,z) in a solid bounded by cylinders?

Calculating the average value of a function in a solid bounded by cylinders can be used in various fields such as physics, engineering, and economics. For example, it can be used to find the average temperature distribution in a cylindrical reactor or to determine the average velocity of a fluid in a cylindrical pipe.

How can I check my calculations for the average value of f(x,y,z) in a solid bounded by cylinders?

You can check your calculations by using multiple methods, such as graphing the function and visually comparing the average value to the shape of the solid, or by using a computer software program to calculate the average value and comparing it to your result. It is always a good idea to double-check your work to ensure accuracy.

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