## 17Calculus - Triple Integrals in Spherical Coordinates

Using Vectors

Applications

### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

### Practice

Practice Problems

Practice Exams

Calculus Tools

### Articles

On this page we cover triple integrals in spherical coordinates and several applications.
Before going through the material on this page, make sure you understand spherical coordinates and how to convert between spherical and rectangular coordinates. See the spherical coordinates page for detailed explanation and practice problems.

Setting Up and Evaluating Triple Integrals in Spherical Coordinates

Okay, so now you know how to convert an equation or a point to spherical coordinates. So how do we set up a triple integral in spherical coordinates? If you are thinking ahead you probably are anticipating that the $$dV$$ term has at least one extra term like we had in cylindrical coordinates and you would be right. The $$dV$$ term in spherical coordinates has two extra terms, $$\rho^2~\sin\phi$$. So $$dV=\rho^2~\sin\phi~d\rho~d\phi~d\theta$$. Remember that the order of $$d\rho~d\phi~d\theta$$ depends on the order of integration and there are six possible orders. This is just one of them.

In integral form, triple integrals in spherical coordinates look like this.

$$\displaystyle{ \iiint\limits_V {f(x,y,z) ~ dV} = }$$ $$\displaystyle{ \iiint\limits_V{ f(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi ) ~ \rho^2~\sin\phi~d\rho~d\phi~d\theta } }$$

VERY IMPORTANT NOTE - - - Do not forget $$\rho^2~\sin\phi$$ in $$\color{red}{\rho^2~\sin\phi}~d\rho~d\phi~d\theta$$ in the above equation. This is the most common mistake made by students learning this technique.

The next step is to describe the volume in spherical coordinates or, in terms of the integral above, determining the limits of integration. The same techniques apply that you used when setting up triple integrals in rectangular coordinates.

Applications of Triple Integrals in Spherical Coordinates

Just as with rectangular and cylindrical coordinates, the meaning of the function f will determine what is being calculated with the triple integral. Here is a review of a couple of examples.

f

$$\iiint_V{f~dV}$$

1

volume

volume density

mass

Also, physical properties like moment of inertia, center of gravity and force can be calculated using triple integrals.

Okay, time for some practice problems.

Practice

Unless otherwise instructed, evaluate these integrals in the spherical coordinate system.

Basic

Evaluate $$\displaystyle{ \iiint\limits_V { x^2 + y^2 + z^2 ~dV } }$$ where V is the unit ball $$x^2 + y^2 + z^2 \leq 1$$, using spherical coordinates.

Problem Statement

Evaluate $$\displaystyle{ \iiint\limits_V { x^2 + y^2 + z^2 ~dV } }$$ where V is the unit ball $$x^2 + y^2 + z^2 \leq 1$$, using spherical coordinates.

$$4 \pi/5$$

Problem Statement

Evaluate $$\displaystyle{ \iiint\limits_V { x^2 + y^2 + z^2 ~dV } }$$ where V is the unit ball $$x^2 + y^2 + z^2 \leq 1$$, using spherical coordinates.

Solution

### 1935 video

video by PatrickJMT

$$4 \pi/5$$

Log in to rate this practice problem and to see it's current rating.

Integrate $$\displaystyle{ f(x,y,z) = 3e^{(x^2 + y^2 + z^2)^{3/2}} }$$ over the region inside $$x^2 + y^2 + z^2 = 1$$ in the first octant, using spherical coordinates.

Problem Statement

Integrate $$\displaystyle{ f(x,y,z) = 3e^{(x^2 + y^2 + z^2)^{3/2}} }$$ over the region inside $$x^2 + y^2 + z^2 = 1$$ in the first octant, using spherical coordinates.

$$\pi(e-1)/2$$

Problem Statement

Integrate $$\displaystyle{ f(x,y,z) = 3e^{(x^2 + y^2 + z^2)^{3/2}} }$$ over the region inside $$x^2 + y^2 + z^2 = 1$$ in the first octant, using spherical coordinates.

Solution

### 1936 video

video by MIP4U

$$\pi(e-1)/2$$

Log in to rate this practice problem and to see it's current rating.

Determine the volume outside $$x^2 + y^2 + z^2 = 1$$ and inside $$x^2 + y^2 + z^2 = 4$$.

Problem Statement

Determine the volume outside $$x^2 + y^2 + z^2 = 1$$ and inside $$x^2 + y^2 + z^2 = 4$$.

$$28 \pi/3$$

Problem Statement

Determine the volume outside $$x^2 + y^2 + z^2 = 1$$ and inside $$x^2 + y^2 + z^2 = 4$$.

Solution

### 1937 video

video by MIP4U

$$28 \pi/3$$

Log in to rate this practice problem and to see it's current rating.

Calculate the volume of the region outside the cone $$\phi = \pi/3$$ and inside the sphere $$\rho = 6 \cos \phi$$.

Problem Statement

Calculate the volume of the region outside the cone $$\phi = \pi/3$$ and inside the sphere $$\rho = 6 \cos \phi$$.

$$9 \pi/4$$

Problem Statement

Calculate the volume of the region outside the cone $$\phi = \pi/3$$ and inside the sphere $$\rho = 6 \cos \phi$$.

Solution

### 1938 video

video by MIP4U

$$9 \pi/4$$

Log in to rate this practice problem and to see it's current rating.

Use spherical coordinates to evaluate $$\iiint_V{ ( x^2 + y^2 + z^2 )^2 ~dV }$$ where V is the ball with center $$(0,0,0)$$ and radius 5.

Problem Statement

Use spherical coordinates to evaluate $$\iiint_V{ ( x^2 + y^2 + z^2 )^2 ~dV }$$ where V is the ball with center $$(0,0,0)$$ and radius 5.

$$312500\pi/7$$

Problem Statement

Use spherical coordinates to evaluate $$\iiint_V{ ( x^2 + y^2 + z^2 )^2 ~dV }$$ where V is the ball with center $$(0,0,0)$$ and radius 5.

Solution

Note: Although she calls this a volume in the video, it really is not a volume. Basically, she is evaluating the function $$( x^2 + y^2 + z^2 )^2$$ over the volume. If she was calculating the volume, the function would be 1.

### 1941 video

video by Krista King Math

$$312500\pi/7$$

Log in to rate this practice problem and to see it's current rating.

Evaluate, using spherical coordinates, $$\displaystyle{ \iiint_V{ \frac{1}{x^2+y^2+z^2}dV } }$$ where the volume V is between the spheres $$x^2 + y^2 + z^2 = 4$$ and $$x^2 + y^2 + z^2 = 25$$ in the first octant.

Problem Statement

Evaluate, using spherical coordinates, $$\displaystyle{ \iiint_V{ \frac{1}{x^2+y^2+z^2}dV } }$$ where the volume V is between the spheres $$x^2 + y^2 + z^2 = 4$$ and $$x^2 + y^2 + z^2 = 25$$ in the first octant.

$$3\pi/2$$

Problem Statement

Evaluate, using spherical coordinates, $$\displaystyle{ \iiint_V{ \frac{1}{x^2+y^2+z^2}dV } }$$ where the volume V is between the spheres $$x^2 + y^2 + z^2 = 4$$ and $$x^2 + y^2 + z^2 = 25$$ in the first octant.

Solution

### 1942 video

video by MIP4U

$$3\pi/2$$

Log in to rate this practice problem and to see it's current rating.

Calculate the volume of a sphere of radius R where $$\phi$$ ranges from 0 to $$\pi/6$$.

Problem Statement

Calculate the volume of a sphere of radius R where $$\phi$$ ranges from 0 to $$\pi/6$$.

Hint

The integral you set up should look like $$V = \iiint dV = \int_0^{\pi/6}{ \int_0^{2\pi}{ \int_0^R{ \rho^2\sin\phi ~ d\rho ~d\theta ~d\phi }}}$$.
Of course, since the limits of integration are constants, you can integrate in any order.

Problem Statement

Calculate the volume of a sphere of radius R where $$\phi$$ ranges from 0 to $$\pi/6$$.

$$\pi R^3 (2-\sqrt{3})/3$$

Problem Statement

Calculate the volume of a sphere of radius R where $$\phi$$ ranges from 0 to $$\pi/6$$.

Hint

The integral you set up should look like $$V = \iiint dV = \int_0^{\pi/6}{ \int_0^{2\pi}{ \int_0^R{ \rho^2\sin\phi ~ d\rho ~d\theta ~d\phi }}}$$.
Of course, since the limits of integration are constants, you can integrate in any order.

Solution

### 2224 video

video by Michel vanBiezen

$$\pi R^3 (2-\sqrt{3})/3$$

Log in to rate this practice problem and to see it's current rating.

Intermediate

Express as a triple integral, using spherical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$ and evaluate.

Problem Statement

Express as a triple integral, using spherical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$ and evaluate.

$$\pi a^3$$

Problem Statement

Express as a triple integral, using spherical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$ and evaluate.

Solution

Note - This same problem is set up in rectangular coordinates on the rectangular coordinates page and in cylindrical coordinates on the cylindrical coordinates page.

### 1933 video

video by Dr Chris Tisdell

$$\pi a^3$$

Log in to rate this practice problem and to see it's current rating.

You CAN Ace Calculus

 spherical coordinates partial integrals triple integrals triple integrals in cylindrical coordinates

### Trig Identities and Formulas

basic trig identities

$$\sin^2\theta+\cos^2\theta=1$$   |   $$1+\tan^2\theta=\sec^2\theta$$

$$\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}$$   |   $$\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}$$

$$\displaystyle{\sec\theta=\frac{1}{\cos\theta}}$$   |   $$\displaystyle{\csc\theta=\frac{1}{\sin\theta}}$$

power reduction (half-angle) formulae

$$\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}$$   |   $$\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}$$

double angle formulae

$$\sin(2\theta)=2\sin\theta\cos\theta$$   |   $$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.
 Setting Up and Evaluating Triple Integrals Applications Practice

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.