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 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

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > partial integrals > triple integrals (spherical)

 Setting Up and Evaluating Triple Integrals Applications Practice

On this page we cover triple integrals in spherical coordinates and several applications. Triple integrals in rectangular coordinates are covered on a separate page, as well as triple integrals in cylindrical coordinates.
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

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - Unless otherwise instructed, evaluate these integrals in the spherical coordinate system. (Some of these problems can be successfully worked in more than one coordinate system.)

Basic Problems

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 solution video

video by PatrickJMT

$$4 \pi/5$$

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 solution video

video by MIP4U

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

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 solution video

video by MIP4U

$$28 \pi/3$$

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 solution video

video by MIP4U

$$9 \pi/4$$

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 solution video

video by Krista King Math

$$312500\pi/7$$

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 solution video

video by MIP4U

$$3\pi/2$$

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 solution video

video by Michel vanBiezen

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

Intermediate Problems

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 solution video

video by Dr Chris Tisdell

$$\pi a^3$$