# 15.1.34 Evaluate triple integral

• MHB
• karush
In summary, the conversation discusses the evaluation of a triple integral with respect to x, y, and z. The final result is $I=2$ and the detail of the dx piece is also shown. There were also discussions about typos and the difficulty of working with trigonometric functions.
karush
Gold Member
MHB
15.1.34 Evaluate
$\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$
integrat dz
$\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy$
integrat dx
$\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$
integrat dy
$I=2$

ok I think its correct, took me an hour to do... trig was confusing
typos mabybe

Show the detail on that dx piece.

karush said:
15.1.34 Evaluate
$\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$
integrat dz
$\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy$
integrat dx$This is correct and it can be written$I=\left(\int_0^{3\pi/2} sin(y)dy\right)\left(\int_0^\pi sin(x)dx\right)= \left[-cos(y)\right]_0^{3\pi/2}\left[-cos(x)\right]_0^\pi= (1)(2)= 2$By the way, in English, "integrate" has an "e" on the end.$\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$integrat dy$I=2\$

ok I think its correct, took me an hour to do... trig was confusing
typos mabybe

## 1. What is a triple integral?

A triple integral is a type of mathematical operation used in multivariable calculus to find the volume of a three-dimensional region. It involves evaluating a function over a three-dimensional region by dividing it into infinitesimal parts and summing them up.

## 2. How do you evaluate a triple integral?

To evaluate a triple integral, you need to determine the limits of integration for each variable, set up the integral using the appropriate order of integration, and then solve the integral using integration techniques such as substitution or integration by parts.

## 3. What is the purpose of evaluating a triple integral?

The purpose of evaluating a triple integral is to find the volume of a three-dimensional region. It is commonly used in physics, engineering, and other fields to calculate the mass, center of mass, and other physical properties of three-dimensional objects.

## 4. What are the different types of triple integrals?

There are two types of triple integrals: Type I and Type II. Type I triple integrals are used when the region of integration is bounded by two or more planes, while Type II triple integrals are used when the region of integration is bounded by a surface and two or more planes.

## 5. What are some common applications of triple integrals?

Triple integrals have many applications in physics, engineering, and other fields. Some common applications include calculating the mass and center of mass of a three-dimensional object, finding the moment of inertia of a solid, and determining the electric charge of a three-dimensional distribution of charge.

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