# Is My Triple Integral Calculation Correct or Is There an Error in the Book?

• mikan
In summary, a triple integral is a mathematical tool used to calculate the volume of a three-dimensional shape by integrating a function over a three-dimensional region. It is commonly used in homework problems involving volume, mass, center of mass, fluid dynamics, and electricity and magnetism. The steps to solve a triple integral include determining limits of integration, setting up the integrand, and evaluating the innermost, next inner, and outermost integrals. A triple integral can have a negative value, indicating that the volume or mass is below the xy-plane. Real-world applications of triple integrals include determining mass and center of mass, analyzing electric and magnetic fields, and calculating probabilities.
mikan
Hi,
my result of

$$\int \int \int_{A} xyz dxdydz$$

where

$$A = \{(x,y,z); x^2+y^2+z^2 \leq 2, x \geq 0, y \geq 0, z \geq 0 \}$$

is

$$\frac {8}{48}$$,

but book says

$$\frac{1}{48}$$.

Is the book right? Could you please verify?

Thank you
Michael

Hey Michael,

I also got 8/48 using:

$$x = \rho \sin\phi \cos\theta, y = \rho \sin\phi \sin\theta, z = \rho \cos\phi$$ with $$0 < \phi < \frac{\pi}{2}, 0 < \theta < \frac{\pi}{2}, 0 < \rho < \sqrt{2}$$

so it looks to be correct, although I'd suggest checking everything one more time just to be sure.

Last edited:

## 1. What is a triple integral?

A triple integral is a mathematical tool used in calculus and multivariable calculus to calculate the volume of a three-dimensional shape. It involves integrating a function over a three-dimensional region, using three different variables.

## 2. When is a triple integral used in homework?

A triple integral is commonly used in homework problems that involve calculating the volume or mass of a three-dimensional object, finding the center of mass, or solving problems related to fluid dynamics or electricity and magnetism.

## 3. What are the steps to solve a triple integral?

The steps to solve a triple integral include: 1) determining the limits of integration for each variable, 2) setting up the integrand (function to be integrated), 3) evaluating the innermost integral, 4) evaluating the next inner integral, and 5) evaluating the outermost integral.

## 4. Can a triple integral have a negative value?

Yes, a triple integral can have a negative value. The value of a triple integral depends on the orientation of the coordinate system and the orientation of the shape being integrated. A negative value indicates that the volume or mass is located below the xy-plane.

## 5. What are some real-world applications of triple integrals?

Triple integrals have numerous real-world applications, including calculating the mass and center of mass of a three-dimensional object, determining the volume of a fluid in a container, analyzing electric and magnetic fields, and calculating probabilities in statistics and economics.

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