Help w/ double integration to solve common volume of two intersecting cylinders

In summary, the conversation revolves around finding the volume bounded by two surfaces, x^2 + y^2 = r^2 and y^2 + z^2 = r^2, using double integrals. The problem involves difficulty in visualizing the integrand and setting up the integral, and the solution is suggested to be done in the order of dx, dy, and dz with specific limits of integration involving constants of r. The suggestion is to then multiply the final answer by 8. One person confirms that this approach yields the correct answer of (16/3)*r^3. There is also a note from the moderator clarifying that calculus questions should be asked in a specific forum.
  • #1
Theelectricchild
260
0
Hi I am taking MV calc and a paticular question in the double integrals chapter asks to find the volume bounded by x^2 + y^2 = r^2 and y^2 +
z^2 = r^2. I already know what the shape looks like (Steinmatic solid) and also know the answer can be achieved using single integration as well, but here I am having difficulty visualizing the integrand for a D. Integral--- the limits of integration will definitely involve constants of r. What would be your suggestion for setting up the integral? The shape is identical on all sides and symmetical--- could there be a way to solve one region and multiply the answer to get the volume, or something along those lines?

Thanks for your help.
 
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  • #2
The integrand is 1. What the limits are is more interesting. do it in the order, what dx then dy then dz

x from 0 to sqrt(r^2-y^2),

y from 0 to sqrt(r^2-z^2)

z from 0 to r

multiply that answer by 8

That sound right to everyone else?
 
  • #3
Multiplying by 8 makes total sense! Thanks i ended up getting (16/3)*r^3 exactly what it should be! Thanks.
 
  • #4
how would u write it if it were a double integral not triple?
 
  • #5

1) What is double integration?

Double integration is a mathematical method used to calculate the volume of a three-dimensional shape by integrating a function over two variables.

2) How is double integration used to solve the volume of two intersecting cylinders?

To solve the volume of two intersecting cylinders using double integration, you would first need to set up the integral by finding the limits of integration for the two variables (usually x and y). Then, you would need to integrate the function representing the cross-sectional area of the intersecting cylinders with respect to both variables to find the total volume.

3) What are the common applications of double integration in science?

Double integration is commonly used in various scientific fields, such as physics, engineering, and mathematics, to calculate volumes, areas, and other physical quantities of three-dimensional objects.

4) Are there any special techniques for solving double integration problems?

Yes, there are several techniques for solving double integration problems, such as using symmetry, changing the order of integration, or using polar or spherical coordinates. It is important to choose the most appropriate technique based on the problem at hand.

5) Are there any limitations or challenges when using double integration to solve volume problems?

One of the main limitations of using double integration to solve volume problems is that it can be a time-consuming and complex process, especially for more complicated shapes. It also requires a good understanding of mathematical concepts and techniques. Additionally, the accuracy of the solution can be affected by any errors in setting up the integral or performing the integration.

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