# Triple integral in spherical coordinates

• hitemup
In summary, the problem asks to evaluate the triple integral ∫∫∫R(x^2+y^2+z^2)dV, where R is a cylinder with bounds 0≤x^2+y^2≤a^2 and 0≤z≤h. The attempt at a solution was to use spherical coordinates, but it was suggested to use cylindrical coordinates instead. The process for setting up the integral in spherical coordinates is explained, and the suggestion to break it up into two integrals is given.
hitemup

## Homework Statement

Evaluate
$$\int \int \int _R (x^2+y^2+z^2)dV$$

where R is the cylinder
$0\leq x^2+y^2\leq a^2$,
$0\leq z\leq h$

## Homework Equations

[/B]
$$x = Rsin\phi cos\theta$$
$$y = Rsin\phi sin\theta$$
$$z = Rcos\phi$$

## The Attempt at a Solution

[/B]
$$2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta \int_{0}^{h/cos\phi}dR R^4sin\phi$$

Are the bounds correct?

Last edited:
Do you have to use spherical co-ordinates, why not cylindrical ones ?

hitemup said:

## Homework Statement

Evaluate
$$\int \int \int _R (x^2+y^2+z^2)dV$$

where R is the cylinder
$0\leq x^2+y^2\leq a^2$,
$0\leq z\leq h$

## Homework Equations

[/B]
$$x = Rsin\phi cos\theta$$
$$y = Rsin\phi sin\theta$$
$$z = Rcos\phi$$

## The Attempt at a Solution

[/B]
$$2*\int_{0}^{\pi/2}d\phi \int_{0}^{2\pi}d\theta \int_{0}^{h/cos\phi}dR R^4sin\phi$$

Are the bounds correct?
This is not set properly if you insist on using spherical coordinates. You would have to break it up into two integrals, one where ##\rho## goes from ##0## to the side of the cylinder and another where it goes from ##0## to the top. A much better choice is cylindrical coordinates. Try setting it up that way and check back with us.

Yes I realized at one time it would be easier if I used cylindrical coordinates, since the region itself is a cylinder. But I just wanted to know how to proceed with spherical coordinates.

Noctisdark
hitemup said:
Yes I realized at one time it would be easier if I used cylindrical coordinates, since the region itself is a cylinder. But I just wanted to know how to proceed with spherical coordinates.

OK, so give it a go. First set up$$\int_{\theta \text{ limits}}\int_{\phi \text{ limits}}\int_0^{\rho \text{ on the sides in terms of } \phi} \text{Integrand } \rho^2\sin\phi~d\rho d\phi d\theta$$then add$$\int_{\theta \text{ limits}}\int_{\phi \text{ limits}}\int_0^{\rho \text{ on the top in terms of } \phi} \text{Integrand }\rho^2\sin\phi~d\rho d\phi d\theta$$

EDIT : Sorry, i thaught they were cylindrical

## 1. What is a triple integral in spherical coordinates?

A triple integral in spherical coordinates is a mathematical tool used to calculate the volume of a three-dimensional object that has spherical symmetry, such as a sphere or a cone.

## 2. How is a triple integral in spherical coordinates different from a regular triple integral?

A triple integral in spherical coordinates takes into account the spherical nature of the object being integrated, while a regular triple integral only considers rectangular coordinates. This means that the limits of integration and the integrand must be expressed in terms of spherical coordinates.

## 3. What is the formula for a triple integral in spherical coordinates?

The formula for a triple integral in spherical coordinates is ∭f(r,θ,φ)r²sin(φ)dV, where r is the radius, θ is the azimuthal angle, φ is the polar angle, and dV is the differential volume element.

## 4. How do you convert a regular triple integral into a triple integral in spherical coordinates?

To convert a regular triple integral into a triple integral in spherical coordinates, you need to express the limits of integration and the integrand in terms of spherical coordinates. This can be done using the following equations: x = r sin(φ)cos(θ), y = r sin(φ)sin(θ), z = r cos(φ), and dV = r²sin(φ)drdθdφ.

## 5. What are the applications of triple integrals in spherical coordinates?

Triple integrals in spherical coordinates are commonly used in physics and engineering to calculate the volume of objects with spherical symmetry, such as planets, stars, and satellites. They are also used in calculus to solve problems involving spherical coordinates, such as finding the mass or center of mass of a three-dimensional object.

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