Solve Volume V of Solid: x^2+y^2+z^2=4 & x^2+y^2=1

  • Thread starter stratusfactio
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In summary, you found the volume V of the solid inside both x^2 + y^2 + z^2 = 4 and x^2 + y^2 = 1. However, you got the wrong answer because you forgot to account for the symmetry.
  • #1
stratusfactio
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Homework Statement



Find the volume V of the solid inside both x^2 + y^2 + z^2 = 4 and x^2 + y^2 = 1.

Homework Equations


So I get how to set it up; you use cylindrical coordinates because it makes life a whole lot simpler BUT the answer is (4pi/3)(8-3^(3/2)) and I got (2pi/3)(8-3^(3/2)). So as you can see, I'm off by a factor of 2.


The Attempt at a Solution


The integrands I have are: z=(0,sqrt(4-r^2); theta = (0, 2pi) and r = (0,1). Since I'm off by a factor of 2, I'm thinking that for theta I should integrate from 0 to 4pi, but conceptually I don't get why.

Any and all help would be much appreciated! :D
 
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  • #2
You just found the volume above the xy plane. There is an equal volume below the xy plane.
 
  • #3
I love this freaking forum! so would theta range from -2pi to 2pi ? I only did with respect to the positive axis. Or even simpler just multiply by 2 because I found the area for just half of the xy plane?
 
  • #4
stratusfactio said:
I love this freaking forum! so would theta range from -2pi to 2pi ? I only did with respect to the positive axis. Or even simpler just multiply by 2 because I found the area for just half of the xy plane?

No about the theta; that isn't where the problem is. The problem is your z limit. You went from z = 0 to z on the top surface. You need to either change your lower limit to z on the bottom surface or double your answer because of the symmetry.
 
  • #5
THANK YOU SO MUCH! I now see hwere I went wrong. I just reattempted the problem using your advice and changed the z limits from (-sqrt(4-r^2),sqrt(4-r^2)) and got the answer I was supposed to get.

Thanks again for your help and speedy response!
 

FAQ: Solve Volume V of Solid: x^2+y^2+z^2=4 & x^2+y^2=1

What is the equation for the solid in this problem?

The equation for the solid is x^2 + y^2 + z^2 = 4, where x, y, and z represent the solid's coordinates in a three-dimensional space.

How can I solve for the volume of this solid?

To solve for the volume of this solid, you can use the method of cylindrical shells, where you integrate the cross-sectional area of the solid with respect to the z-axis. You can also use triple integrals in cylindrical or spherical coordinates to calculate the volume.

3. What are the boundaries for the integrals when solving for the volume?

The boundaries for the integrals will depend on the method you choose to solve for the volume. For cylindrical shells, the boundaries for the integral will be from 0 to 2π for the angle, from 0 to 1 for the radius, and from 0 to √(4 - z^2) for the height. For triple integrals, the boundaries will depend on the chosen coordinates, but will generally be defined by the equations in the given solid.

4. Can I use software to solve for the volume of this solid?

Yes, you can use mathematical software such as Wolfram Alpha or Matlab to solve for the volume of this solid. However, it is important to understand the mathematical concepts and equations involved in order to properly interpret the results.

5. How can I verify my solution for the volume of this solid?

You can verify your solution by using different methods to solve for the volume, such as using cylindrical shells or triple integrals in different coordinate systems. You can also check your solution using visualization software or by comparing it to known volumes of similar solids.

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