# Triple Integral - Volume

• Larrytsai
In summary, a triple integral is a mathematical concept used to calculate the volume of a three-dimensional shape by integrating a function over a three-dimensional region within a coordinate system. It differs from a regular integral, which calculates the area under a curve in a two-dimensional plane. A triple integral is calculated by integrating a function over each variable in a three-dimensional coordinate system, and is commonly used in physics, engineering, and other scientific fields to solve problems involving three-dimensional shapes and volumes. It can be used for any three-dimensional shape as long as it can be defined within a three-dimensional coordinate system and the function to be integrated is defined over the entire region.
Larrytsai

## Homework Statement

Evaluate the triple integral of (x+5y)dV where E is bounded by the parabolic cylinder y=3x^2 and the planes z=9x, y=18x and z=0.

## The Attempt at a Solution

My solution is this...

27*6^5 /5 - 162*6^4 /4 + (45/2)*(9*6^6 /6 - 324*6^4 /4) = -797817.6

My steps are here:

Bounds:
0<= z<=9x
18x<= y <=3x^2
0<= x<=6

(x+5y)dzdydx
(x+5y)z
(9yx^2 + 45y^2x /2)dx
[27x^4 - 162x^3 + (45/2)*(9x^5 - 324x^3)]dx

EVALUATED FROM 0 TO 6
27x^5 /5 - 162x^4 /4 + (45/2)*(9x^6 /6 - 324x^4 /4)

## What is a triple integral?

A triple integral is a mathematical concept used to calculate the volume of a three-dimensional shape. It involves integrating a function over a three-dimensional region within a coordinate system.

## What is the difference between a triple integral and a regular integral?

A regular integral calculates the area under a curve in a two-dimensional plane, while a triple integral calculates the volume of a three-dimensional shape.

## How is a triple integral calculated?

A triple integral is calculated by integrating a function over each variable in a three-dimensional coordinate system. This means that the integral will have three variables, and the limits of integration will correspond to the bounds of the three-dimensional region.

## What is the purpose of using a triple integral?

The main purpose of a triple integral is to calculate the volume of a three-dimensional object. It is commonly used in physics, engineering, and other scientific fields to solve problems involving three-dimensional shapes and volumes.

## Can a triple integral be used for any shape?

Yes, a triple integral can be used to find the volume of any three-dimensional shape, as long as the shape can be defined within a three-dimensional coordinate system and the function to be integrated is defined over the entire region.

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