Volume of a Solid using Triple Integrals

In summary, the conversation discusses using a triple integral to find the volume of a solid enclosed by a cylinder and three planes. The integral is set up in rectangular coordinates with specific limits, leading to a final answer of 512. The speaker seeks confirmation on the correctness of their work.
  • #1
jualin
8
0

Homework Statement



Use a triple integral to find the volume of the solid enclosed by the cylinder z=y2 and the planes x=0, x=6, and z=16. Set up the integral in rectangular coordinates and work it out in any coordinates.

Homework Equations


The Attempt at a Solution



I set up the triple integral using these orders of integration:

D = { (x,y,z) | 0 < x < 6, -4 < y < 4, y2 < z < 16 }

And I obtained an answer of 512. I just wanted to make sure that I did all the work correctly, and that someone more experienced could say if it is correct.
Thank you!
 
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  • #2
Your limits look good so if you didn't make any arithmetic mistakes it should be correct.
 
  • #3
LCKurtz said:
Your limits look good so if you didn't make any arithmetic mistakes it should be correct.
Thank you very much, Mr Kurtz!
 

FAQ: Volume of a Solid using Triple Integrals

What is the concept of volume in triple integrals?

Volume in triple integrals refers to the measure of space occupied by a solid object in three-dimensional space. It is calculated by dividing the object into infinitesimally small pieces and adding up the volume of each piece using triple integrals.

How is the volume of a solid determined using triple integrals?

The volume of a solid is determined by setting up a triple integral that integrates the infinitesimal volumes of the solid along the three axes of a three-dimensional coordinate system. These infinitesimal volumes are calculated using the height, width, and depth of the solid at a given point.

What is the difference between a single integral and a triple integral?

A single integral calculates the area under a curve on a two-dimensional plane, while a triple integral calculates the volume of a solid in three-dimensional space. Triple integrals require three variables and are used to integrate over a three-dimensional region, while single integrals only require one variable and are used to integrate over a two-dimensional region.

How do you set up a triple integral for a given solid?

To set up a triple integral for a given solid, you first need to determine the limits of integration for each variable. This involves finding the bounds of the solid along each axis. Then, you need to determine the function that represents the infinitesimal volume of the solid at a given point. Finally, you can set up the triple integral by integrating the function over the appropriate limits of integration for each variable.

What are some practical applications of triple integrals in science?

Triple integrals have numerous practical applications in science, particularly in physics and engineering. They are used to calculate the volume of objects such as tanks, pipes, and buildings. They can also be used to find the center of mass for irregularly shaped objects, or to calculate the moment of inertia for rotating bodies. Additionally, triple integrals are used in fluid mechanics to calculate the flow of fluids through three-dimensional channels and pipes.

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