# Triple Integral: Volume of a Solid

• iamalexalright
In summary, the conversation discusses using a triple integral to find the volume of a solid enclosed by a cylinder and two planes. The bounds of the integral are discussed and the correct solution is determined to be 135*pi.
iamalexalright

## Homework Statement

Been awhile since I've done them and my memory/reasoning isn't so great apparently...

Use the triple integral to find the volume of the given solid:
The solid enclosed by the cylinder
$$x^{2} + y^{2} = 9$$
and the planes y + z = 16 and z = 1. 2. The attempt at a solution
Difficulty is always setting up the bounds of the integral...
$$-3 \leq y \leq 3$$
$$1 \leq z \leq 16 - y$$
having problems with the xwould it be:
$$-\sqrt{9 - y^{2}} \leq x \leq \sqrt{9 - y^{2}}$$ ?

Sure. The planes don't intersect inside the cylinder. So you can parametrize the integral over the x,y in the circle defining the cylinder without worrying about the z value. If the planes had intersected inside the circle they would have had to give you a more elaborate description of the region.

135*pi, cool! Thanks Dick

iamalexalright said:
135*pi, cool! Thanks Dick

That's what I get. :)

## What is a triple integral?

A triple integral is a mathematical tool used to calculate the volume of a three-dimensional solid. It involves integrating a function over a three-dimensional region in space.

## How do you set up a triple integral?

To set up a triple integral, you need to define the limits of integration for each variable (x, y, and z) and the function to be integrated. This typically involves breaking the solid into smaller, simpler shapes and determining the range of each variable within those shapes.

## What is the difference between a single, double, and triple integral?

A single integral is used to find the area under a curve, a double integral is used to find the volume under a surface, and a triple integral is used to find the volume of a three-dimensional solid. In essence, each type of integral adds another dimension to the calculation.

## What are some real-life applications of triple integrals?

Triple integrals are commonly used in physics and engineering to calculate the moment of inertia of a three-dimensional object, the mass of a three-dimensional shape, and the center of mass of a three-dimensional object. They are also used in economics to calculate the volume of a three-dimensional production or consumption region.

## Can a triple integral be used to calculate the volume of any solid?

Yes, a triple integral can be used to calculate the volume of any solid, as long as the shape of the solid can be expressed as a function or a combination of functions in three dimensions.

Replies
14
Views
1K
Replies
6
Views
1K
Replies
9
Views
1K
Replies
27
Views
2K
Replies
3
Views
1K
Replies
11
Views
2K
Replies
4
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
3
Views
1K