Finding a volume by triple integration

In summary, the conversation was about finding the volume between a cone and a sphere in the first octant. The person used cylindrical coordinates and found the limits to be 0<theta<pi/2, 0<r<1/sqrt(2), and r<z<sqrt(1-r^2). They also did a triple integral and obtained the answer of \frac{\pi}{6} - \frac{\sqrt{2}\pi}{12}. They asked for confirmation on the correctness of their limits and suggested trying spherical coordinates for easier limits.
  • #1
Juggler123
83
0
I need to find the volume between the cone z=sqrt(x^2+y^2) and the sphere x^2+y^2+z^2=1 that lies in the first octant. Now I've used cylindrical coordinates for this and found the limits to be

0<theta<pi/2
0<r<1/sqrt(2)
r<z<sqrt(1-r^2)

I've done the triple integral and found the answer to be [tex]\frac{\pi}{6}[/tex] - [tex]\frac{\sqrt{2}\pi}{12}[/tex]

Just want to check to see if this is correct, as I'm not fully sure about the limits I've got. Any help would be great, thanks!
 
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  • #2
Your limits look OK. I didn't check your answer though. You might try setting it up in spherical coordinates where you will find the limits are easier. And if you get the same answer, you're good to go.
 

1. How is the volume of a solid determined using triple integration?

The volume of a solid can be determined using triple integration by dividing the solid into infinitesimally small pieces and integrating over the entire volume using three variables, typically x, y, and z. This process involves finding the limits of integration for each variable and integrating the function representing the solid over those limits.

2. What is the difference between single, double, and triple integration?

Single integration involves integrating a function over one variable, double integration involves integrating a function over two variables, and triple integration involves integrating a function over three variables. In terms of finding volume, single integration would be used for finding the area under a curve, double integration would be used for finding the volume of a solid with a flat base, and triple integration would be used for finding the volume of a more complex solid.

3. Can triple integration be used for finding the volume of any solid?

Yes, triple integration can be used for finding the volume of any solid, as long as the function representing the solid is integrable and the limits of integration can be determined for each variable.

4. What are some applications of triple integration in real life?

Triple integration has various applications in fields such as physics, engineering, and economics. It can be used to find the volume of objects in 3D space, calculate the mass of an object with varying density, determine the center of mass of an object, and solve optimization problems.

5. Are there any alternative methods for finding volume besides triple integration?

Yes, there are alternative methods for finding volume, such as using geometric formulas for simple shapes or using other integration techniques such as cylindrical or spherical coordinates. However, triple integration is the most general and versatile method for finding volume of any solid in 3D space.

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