\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Triple Integrals in Spherical Coordinates

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

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.

Learn to Remember: Practical Techniques and Exercises to Improve Your Memory

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.

Final Answer

\( 4 \pi/5 \)

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

PatrickJMT - 1935 video solution

video by PatrickJMT

Final Answer

\( 4 \pi/5 \)

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.

Final Answer

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

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

MIP4U - 1936 video solution

video by MIP4U

Final Answer

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

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 \( x^2 + y^2 + z^2 = 4 \).

Final Answer

\( 28 \pi/3 \)

Problem Statement

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

Solution

MIP4U - 1937 video solution

video by MIP4U

Final Answer

\( 28 \pi/3 \)

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 \( \rho = 6 \cos \phi \).

Final Answer

\( 9 \pi/4 \)

Problem Statement

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

Solution

MIP4U - 1938 video solution

video by MIP4U

Final Answer

\( 9 \pi/4 \)

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.

Final Answer

\( 312500\pi/7 \)

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.

Krista King Math - 1941 video solution

video by Krista King Math

Final Answer

\( 312500\pi/7 \)

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.

Final Answer

\( 3\pi/2 \)

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

MIP4U - 1942 video solution

video by MIP4U

Final Answer

\( 3\pi/2 \)

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 \( \pi/6 \).

Final Answer

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

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

Michel vanBiezen - 2224 video solution

video by Michel vanBiezen

Final Answer

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

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.

Final Answer

\( \pi a^3 \)

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.

Dr Chris Tisdell - 1933 video solution

video by Dr Chris Tisdell

Final Answer

\( \pi a^3 \)

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

Really UNDERSTAND Calculus

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

Topics You Need To Understand For This Page

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

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

links

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Related Topics and Links

To bookmark this page and practice problems, log in to your account or set up a free account.

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.

learning and study techniques

Shop Amazon - Used Textbooks - Save up to 90%

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Setting Up and Evaluating Triple Integrals

Applications

Practice

Practice Search

Practice Instructions

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

Do NOT follow this link or you will be banned from the site!

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.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics