13.5 Triple Integrals in Cylindrical and Spherical...

13.5 Triple Integrals in Cylindrical and Spherical Coordinates

In this section we consider again triple integrals∫∫∫

D

f (x , y , z) dV .

Just as it was useful to evaluate certain double integrals using polar coordinates, we’llfind it useful to evaluate certain triple integrals using other polar coordinate systems fordescribing points in space.

We study two such systems:1 the cylindrical coordinate system2 the spherical polar coordinate system

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Cylindrical Coordinates

Cylindrical Coordinates of a point P in space

Let P = (x , y , z) be a point in space described here using rectangularcoordinates.The cylindrical coordinates of P are (r , θ, z), where

(r , θ) are the two-dimensional polar coordinates of the point (x , y),z is the same z coordinate as in the rectangular coordinates.

Thus we can view (r , θ) as being obtained by orthogonally projecting P onto thexy -plane to obtain point Q in the xy -plane and taking the polar coordinates of theprojected point Q.

Exercise.Make a sketch of a point P in a three-dimensional coordinate system and identify on the sketch each of thecylindrical coordinates (r , θ, z).

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Converting between rectangular and cylindrical coordinates

The formulas we will require arex2 + y 2 = r 2

x = r cos θ, y = r sin θ, z = zThe range of values we will need in doing triple integrals with cylindricalcoordinates are as follows:

0 ≤ r <∞−π ≤ θ ≤ π or 0 ≤ θ ≤ 2π−∞ < z <∞

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Exercise.Identify by sketch what are the level surfaces of each of r , θ, and z .

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Triple integrals in cylindrical coordinates

The aim is to write∫∫∫

D

f (x , y , z) dV using cylindrical coordinates.

Just as for using polar coordinates to rewrite a double integral, we have three things todeal with:

Steps in writing∫∫∫

D

f (x , y , z)dV using cylindrical coordinates

Step I: Rewrite the integrand f (x , y , z) in terms of cylindrical coordinatesStep II: Modify the limits of integrationStep III: Rewrite dV as a dz dr dθ multiplied by an additional factor (to bedetermined) which we call the Jacobian factor.

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The way to handle Step I is:replace any terms x2 + y 2 by r 2,replace any other occurrrence of x by r sin θ and y by r sin θ,leave any term involving z alone (since z is a cylindrical coordinate).

The way to handle Step II has to do with the specific types of solids D to which we canapply the method.The types of such solids D turns out to be pretty much the same as for ordinary tripleintegrals in rectangular coordinates. The only difference is that the upper and lowersurfaces can be described easily using polar coordinates

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Types of solids D over which we can integrate in cylindrical coordinates

The form of D we’re looking for consists of three things:A region R in the xy -plane which is easily described in terms oftwo-dimensional polar coordinates (as in section 13.3);A surface we call the “lower surface” given by z = g1(r , θ) for (r , θ) the polarcoordinates of points in R ;A surface we call the “upper surface” given by z = g1(r , θ) for (r , θ) thepolar coordinates of points in R .

Then the solid D is the region that lies between the lower surface and the uppersurface.

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Exercise.Make a sketch which describes the above regions, clearly indicating the solid D, theorthogonal projection R of D, as well as the upper and lower surfaces.

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The third and final step in setting up the triple integral is to decide what is the correctJacobian factor under the integral sign.It is decided by considering the grid of “boxes” created by the level surfaces of each ofthe cylindrical coordinates r , θ, and z .Consider such a “box” where the individual coordinates vary by dr , dθ, dz .If we slice through such a box with the level surface z = c , we get the two-dimensionalbox created by polar coordinates (r , θ), and by section 13.3, this contributes an arear dr dθ. The height of this box if dz , so this gives us the following result:

Jacobian factor for triple integrals in cylindrical coordinatesThe additional Jacobian factor needed is r . Thus in the triple integral we should replace dVwith

dV = r dr dθ dz .

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If we combine the three steps we’ve deduced above for setting up the integral, theresult is


Assuming that D can be described as given above, the triple integral∫∫∫D

f (x , y , z) dV can be written as an iterated triple integral in cylindrical co-

ordinates as follows:∫∫∫D

f (x , y , z) dV =∫ β

α

∫ h2(θ)

h1(θ)

∫ g2(r ,θ)

g1(r ,θ)F (r , θ, z) r dz dr dθ,

whereF (r , θ, z) is the description of f written in cylindrical coordinates,the region R which is the projection of D onto the xy -plane can be describedin polar coordinates by

α ≤ θ ≤ βh1(θ) ≤ r ≤ h2(θ).

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A few comments on the above triple integralThe inner integral with respect to z is done by fixing r and θ and integrating z fromthe lower surface to the upper surface. Once it is done, all of the z ’s will be gone.Don’t forget to include the Jacobian factor of r in this integral.The outer double integral in r and θ is just a regular double integral in polarcoordinates over the region in the xy -plane which is the projection of the solid Dstraight down onto the xy -plane.

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Triple integrals in cylindrical coordinates: Exercises

Exercises.Let R be the region in the xy -plane in the first quadrant lying above the line y = x andbelow the circle x2 + y 2 = 1. Consider

∫∫∫D

e−x2−y2 dV , where D is the region lying above

the region R and below the plane z = 4.a Sketch the region R . Identify the boundary curves of R using cylindrical coordinates.b Write down the above triple integral as an iterated triple integral using rectangular

coordinates.c Now write the above triple integral as an iterated triple integral using cylindrical

coordinates.

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Exercise.

Consider the iterated triple integral∫ 4

−4

∫ √16−x2

−√

16−x2

∫ 4√

x2+y2

11 + x2 + y 2 dz dy dx given in

rectangular coordinates. Let D denote the solid over which we are integrating.a Identify by sketch the associated region R , the lower surface, and the upper surface.b Write the above integral using cylindrical coordinates.

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Exercise.Write down the iterated triple integral in cylindrical coordinates for each of the following.For each, identify the associated region R in the xy -plane and the associated lower andupper surfaces.

a The volume of the solid lying between the cylinders x2 + y 2 = 1 and x2 + y 2 = 4,bounded below by the plane z = 0 and above by the plane z = 4− x − y .

b The average value of f (x , y , z) = x2y 2z for (x , y , z) varying over the solid D boundedbelow by the paraboloid z = x2 + y 2 and above by the plane z = 7.

c The volume of the solid lying above the paraboloid z = x2 + y 2 and below the spherex2 + y 2 + z2 = 50.

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Spherical Coordinates


Let P be a point in space with rectangular coordinates (x , y , z). Let−→OP be the

position vector joining the origin to the point P.The spherical coordinates are named (ρ, φ, θ). They are defined as follows:

ρ = |−→OP| = the distance of P to the origin;

φ = the angle between ~k and−→OP = the angle between the positive z-axis

and−→OP;

θ is the same as for cylindrical coordinates.

Exercise.Make a sketch of a point P in a three-dimensional coordinate system and identify on thesketch each of the spherical coordinates (ρ, θ, z).

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Exercise.Identify both by name and by sketch the level surfaces of each of ρ, φ, and θ.

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Exercise (Converting from spherical to rectangular coordinates)Show how to write each of x , y , and z in terms of the spherical coordinates ρ, φ, and θ.

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We note below the conversions we will need to know how to do that take us betweenrectangular and spherical coordinates.The ones from spherical to rectangular were deduced in the exercise on the previousslide.

Converting between rectangular and spherical coordinates

The formulas we will require arex2 + y 2 + z2 = ρ2

x = ρ sinφ cos θy = ρ sinφ sin θz = ρ cosφ.

The range of values we will need in doing triple integrals with cylindricalcoordinates are as follows:

0 ≤ ρ <∞0 ≤ φ ≤ π−π ≤ θ ≤ π or 0 ≤ θ ≤ 2π

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Triple Integrals in Spherical Coordinates

We wish now to understand how to rewrite∫∫∫

D

f (x , y , z) dV in terms of spherical

coordinates.As before we have three things to take care of:

Steps in writing∫∫∫

D

f (x , y , z)dV using spherical coordinates

Step I: Rewrite the integrand f (x , y , z) in terms of spherical coordinatesStep II: Modify the limits of integrationStep III: Rewrite dV as a dρ dφ dθ multiplied by an additional factor, whichwe call the Jacobian factor.

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Step I is done by making use of the conversions of the previous slide:Always begin by replacing any occurrence of x2 + y 2 + z2 with ρ2;For any other occurrence of x , y , or z make use of the formulas x = ρ sinφ cos θ,y = ρ sinφ sin θ, z = ρ cosφ.

The answer to Step III is that the

Jacobian factor is ρ2 sinφ

This is a bit complicated, more difficult than for cylindrical coordinates, so we’ll discussthe reasons for this later on.

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Step II requires that the solid D be of a certain special type.The analogue of Step II for both rectangular coordinates and cylindrical coordinates,required that D be the solid between an “upper surface” and a “lower surface”.For spherical coordinates, we instead require that D be the solid between an “innersurface” and an “outer surface”.This is because ρ measures distance back to the origin, not a height up or downwhich is what z measures.

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Types of solids D over which we can integrate in spherical coordinates

The solids D we can handle will always be of the following type:φ and θ will always vary independently over a biggest and a smallest value:

φ1 ≤ φ ≤ φ2θ1 ≤ θ ≤ θ2

A surface we call the “inner surface” given by ρ = g1(φ, θ) for (φ, θ) varyingindependently over the above constant values;A surface we call the “outer surface” given by ρ = g2(φ, θ) for (φ, θ) varyingindependently over the above constant values.

Then the solid D is the region that lies between the inner surface and the outersurface.

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Using the results of the last few slides, we can write down the formula we’ve beenlooking for:

Triple integrals in spherical coordinates

Assuming that D can be described as on the previous slide and using the notationon that slide, the triple integral

∫∫∫D

f (x , y , z) dV can be written as an iterated

triple integral in spherical coordinates as follows:∫∫∫D

f (x , y , z) dV =∫ θ2

θ1

∫ φ2

φ1

∫ g2(φ,θ)

g1(φ,θ)F (ρ, φ, θ) ρ2 sinφ dρ dφ dθ,

where F (ρ, φ, θ) is the description of f written in spherical coordinates.

Before we do some exercises, we discuss where the Jacobian factor ρ2 sinφ comes from.

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Triple Integrals in Spherical Coordinates: the Jacobian factor ρ2 sinφ

We put this formula in a broader context: we describe (without any discussion of proof) a generalchange of variables formula that works for all triple integrals.

Suppose we have a triple integral∫∫∫

D

f (x , y , z) dV and wish to write it in terms of some other

variables (u, v ,w) for which we have formulas writing (x , y , z) in terms of (u, v ,w):x = x(u, v ,w)y = y(u, v ,w)z = z(u, v ,w)

With the transformation (u, v ,w) 7→ (x , y , z) we can associate a quantity known as the Jacobian of

the transformation, denoted by ∂(x , y , z)∂(u, v ,w) :

∂(x , y , z)∂(u, v ,w) :=

∣∣∣∣∣∣∣∣∣∣

∂x∂u

∂y∂u

∂z∂u

∂x∂v

∂y∂v

∂z∂v

∂x∂w

∂y∂w

∂z∂w

∣∣∣∣∣∣∣∣∣∣where the absolute value denote the absolute value of the determinant of the given matrix.Then the general change of variables formula is given on the next slide.

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Triple Integrals in Spherical Coordinates: the Jacobian factor ρ2 sinφ

General Change of Variable in Triple Integrals

∫∫∫D

f (x , y , z)dV =∫∫∫

D′

f(

x(u, v ,w), y(u, v ,w), z(u, v ,w))∂(x , y , z)∂(u, v ,w)du dv dw

where D′ is the region D described in terms of the variables u, v , and w .

Brief explanation

The determinant represents the volume of the “box” having the rows of the matrix as the sidevectors of the box.One of the ingredients in a triple integral is a partition of the solid by boxes.If we take each of the “uvw -boxes” in a partition of D′ and map that box via thetransformation (u, v ,w) 7→ (x , y , z), we get a partition of D, but no longer by ordinary boxes,but let’s still call them xyz-boxes.The Jacobian determinant ∂(x ,y ,z)

∂(u,v ,w) gives a measure of the factor by which the volume of thexyz-box changes from the volume of the original uvw -box.The first derivatives arise since we are estimating the shape of the xyz-box using tangentvectors.

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Triple Integrals in Spherical Coordinates: the Jacobian factor ρ2 sinφThen using the above change of variables result, we can confirm what is the Jacobian factor inswitching from rectangular to spherical coordinates. We do this in the next exercise.

Exercise.Show that the Jacobian of the transformation

x = ρ sinφ cos θy = ρ sinφ sin θz = ρ cosφ

is given by∂(x , y , z)∂(ρ, φ, θ) = ρ2 sinφ.

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Triple Integrals in Spherical Coordinates: ExercisesExercise.This question concerns describing certain objects using spherical coordinates.

a Describe the cone z =√

x2 + y 2 using spherical coordinates.b Describe the plane z = 7 using spherical coordinates.c Explain why the equation ρ = 4 cosφ represents a sphere. Do it by converting back to

rectangular coordinates. What are the center and radius of the sphere? Sketch thissphere.

d How would you represent the cylinder r = 7 using spherical coordinates?

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Triple Integrals in Spherical Coordinates: Exercises

Exercise.As you probably know, the volume of a ball of radius R is (4/3)πR3. Use a triple integral inspherical coordinates to confirm this formula.

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Exercise.Write down iterated integrals in spherical coordinates that represent each of the following.For each one, indicate which is the “inner surface” and “outer surface”.

a The volume of the part of the ball x2 + y 2 + z2 ≤ 16 that lies above the plane z = 2.See problem (b) above.

b The volume of the first octant part of the solid lying outside the cone φ = π/4 and insidethe sphere ρ = 4 cosφ.

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Exercise (continuation).Write down iterated integrals in spherical coordinates that represent each of the following.For each one, indicate which is the “inner surface” and “outer surface”.

c The volume of the region between the cylinders r = 1 and r = 2 that is outside the coneφ = π/6 and inside the cone φ = π/3.

d The volume of the region inside the cone z =√

x2 + y 2 that lies between the planesz = 1 and z = 2.

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Exercise.

Evaluate the triple integral∫∫∫

D

(x2 + y 2 + z2)5/2 dV , where D is the unit ball

x2 + y 2 + z2 ≤ 1.

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13.5 Triple Integrals in Cylindrical and Spherical...

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