worksheet 3.4: triple integrals in cylindrical...

Boise State Math 275 (Ultman)

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates

From the Toolbox (what you need from previous classes)

� Know what the volume element dV represents.

� Be able to find limits of integration for double integrals in polar coordinates, and triple integrals

in Cartesian coordinates.

Goals

In this worksheet, you will:

� Use the cylindrical change of coordinate functions to convert expressions in Cartesian coordi-

nates to equations in cylindrical coordinates.

� Set up and evaluate triple integrals in cylindrical coordinates. This includes finding limits of

integration, converting the integrand from Cartesian to cylindrical coordinates, and using the

cylindrical volume element.

Warm-Up: Cylindrical Volume Element dV

The volume element dV in cylindrical coordinates is the volume of an infinitesimal box, where

the base of the box is the polar area element dA, and the height of the box is an infinitesimal

change in z . Under the infinite magnifying glass:

So, the cylindrical volume element is:

dV = dApolar × dz = d d dz

Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates 1

Model 1: Grid Surfaces in Cartesian & Cylindrical Coordinates

Diagram 1A: Grid Surfaces in Cartesian Coordinates (((x, y , z)))

−−−∞∞∞ < x <∞∞∞, −−−∞∞∞ < y <∞∞∞, −−−∞∞∞ < z <∞∞∞

Grid Surfaces:

x === (constant)

(y & z vary)

Grid Surfaces:

y === (constant)

(x & z vary)

Grid Surfaces:

z === (constant)

(x & y vary)

Diagram 1B: Grid Surfaces in Cylindrical Coordinates (((r, θ, z)))

000 ≤≤≤ r <∞∞∞, ω ≤≤≤ θ < ω +++ 222π, −−−∞∞∞ < z <∞∞∞

Grid Surfaces:

r === (constant)

(θ & z vary)

Grid Surfaces:

θ === (constant)

(r & z vary)

Grid Surfaces:

z === (constant)

(r & θ vary)

Critical Thinking Questions

In this section, you will compare grid surfaces in Cartesian and cylindrical coordinates, and

compare polar and cylindrical coordinates.

A grid surface of a 3-d coordinate system is a surface generated by holding one of the coordinates

constant while letting the other two vary.


(Q1) Refer to Diagrams 1A & 1B: Determine whether the following surfaces are grid surfaces in

either Cartesian or cylindrical coordinates. If they are, indicate for which coordinate system(s),

and which coordinate is held constant.

(a) Planes parallel to the yz-plane:

Cartesian: x / y / z held constant / cylindrical: r / θ / z held constant / neither

(b) Planes parallel to the xz-plane:


(c) Planes parallel to the xy -plane:


(d) Cylinders centered about the z-axis:


(e) Half-planes perpendicular to the xy -plane:


(f) Spheres centered about the origin:


(Q2) Refer to Diagrams 1A & 1B: Which of the following solid 3-d regions are bounded by the grid

surfaces of either the Cartesian or cylindrical coordinate system?

(a) Rectangular boxes, with sides parallel to the xyz-coordinate planes:

Cartesian coordinate system / cylindrical coordinate system / neither

(b) Solid cylinders centered about the z-axis:


(c) Solid balls centered about the origin:


(Q3) We will say that a “cylindrical box” is a region bounded by pairs of cylindrical grid surfaces

r = c1, r = c2, θ = k1, θ = k2, z = d1, z = d2 (for c1, c2, k1, k2, d1, d2 constants).

Sketch three different “cylindrical boxes” (refer to Diagram 1B for the bounding grid surfaces):


(Q4) The diagrams below represents a plane parallel to the xy -plane; in fact, this is a grid surface

for cylindrical coordinates, z = (constant) (shown on the right of diagram 1B).

(a) On the left, sketch the curves of intersection for one of the planes z = (constant) with

the set of grid surfaces r = (constant) (shown on the left of Diagram 1B).

(b) In the middle, sketch the curves of intersection for one of the planes z = (constant) with

the set of grid surfaces θ = (constant) (shown on the middle of Diagram 1B).

(c) On the right, combine the curves of intersection from parts (a) and (b).

Intersection of z = c & grid

surfaces r = (constant).


surfaces θ = (constant).


surfaces r, θ = (constant).

(Q5) The curves of intersection you sketched in (Q4) should look familiar. Where have you seen

them before?

(⊕ Q6) The cylindrical coordinate functions x = r cos θ, y = r sin θ, z = z lead to the grid surfaces

you’ve been working with so far. In particular, when r = (constant), the grid surfaces are

cylinders centered about the z-axis.

(a) Find the three coordinate functions so that the cylinders are centered about the x-axis.

x =

y =

z =

(b) Find the three coordinate functions so that the cylinders are centered about the y -axis.

x =

y =

z =


Model 2: Integrating Over Basic Regions

Diagram 2A

Cartesian Coordinates Cylindrical Coordinates

x2 + y 2 ≤ 4 ≤ r ≤

≤ θ ≤

0 ≤ z ≤ 3 ≤ z ≤

Diagram 2B


1 ≤ x2 + y 2 ≤ 4 ≤ r ≤

≤ θ ≤

0 ≤ z ≤ 3 ≤ z ≤

Diagram 2C


1 ≤ x2 + y 2 ≤ 4 ≤ r ≤

x, y ≥ 0 ≤ θ ≤

0 ≤ z ≤ 3 ≤ z ≤


In this section, you will work with triple integrals in cylindrical coordinates for regions bounded

by cylindrical grid surfaces.

(Q7) In Diagrams 2A, 2B & 2C: Use the equations in Cartesian coordinates to find the equations

for the bounding surfaces in cylindrical coordinates. Label the bounding surfaces in the three

diagrams with their equations in cylindrical coordinates.


(Q8) Use the limits of integration to match the triple integrals below with its region of integration

(one of the regions in Diagrams 2A, 2B, or 2C).

ˆ 30

ˆ 2π0

ˆ 21

(r + z) r dr dθ dz Region from Diagram 2

ˆ 30

ˆ 2π0

ˆ 20

5z r dr dθ dz Region from Diagram 2

ˆ 30

ˆ π/2

0

ˆ 21

z sin θ r dr dθ dz Region from Diagram 2

ˆ π/2

0

ˆ 21

ˆ 30

z2 r dz dr dθ Region from Diagram 2

(Q9) Sketch the solid cylinder W bounded by the cylinder x2 + y 2 = R2, between the xy -plane and

the plane z = H.

(Q10) (a) Write down a triple integral in cylindrical coordinates that gives the volume of the solid

cylinder W from (Q9). Make sure you use the cylindrical volume element dV that you

found in the Warm-Up! Then, evaluate the integral to show that the volume of this

region is VW = πR2H.

VW =

˚W

1 dV =

˚W

dV =

ˆ ˆ ˆd d d

(b) What is the difference between the lower-case r and the upper-case R?


(⊕ Q11) Which of the following represent the volume of the solid cylinder W from (Q9)? Circle all

correct answers, then “fix” the incorrect answers.

(a)

ˆ 2π0

ˆ H

0

ˆ R

0

dr dz dθ

(b)

ˆ 2π0

ˆ H

0

ˆ R

0

Rdr dz dθ

(c) 2

ˆ π

0

ˆ H

0

ˆ R

0

r dr dz dθ

(d)

ˆ 2π0

ˆ R

0

ˆ H

0

r dr dz dθ

(e) 2

ˆ 2π0

ˆ H/2

0

ˆ R

0

r dr dz dθ

(f) None of the above represent the volume of the region W .

(⊕ Q12) Suppose W is the solid region bounded by the cylinder x2 + y 2 = 4 and the planes z = 0 and

z = x + y + 3. Set up a triple integral in cylindrical coordinates that gives the volume of W .

(⊕ Q13) Suppose W is the solid cylinder bounded by the cylinder x2 + y 2 = 25 and the planes z = 0

and z = 3. Find the value of the radius r that separates the solid cylinder into two regions of

equal volume.


Model 3: Integrating Over Other Regions

Diagram 3A: Region inside the cone z ===√x2 + y 2 and below the plane z === 333

In cylindrical coordinates, the equations of the cone and plane are:

cone: z = r plane: z = 3

Diagram 3B: Region inside the sphere x222 +++ y 222 +++ z222 === 999

In cylindrical coordinates, the equation of the sphere is:

sphere: r 2 + z2 = 9


In this section, you will work with triple integrals in cylindrical coordinates for regions bounded

by surfaces that are not grid surfaces.


(Q14) Using the sketches in Diagrams 3A and 3B, match the given limits of integration for integrals

over the cone in Diagram 3A and the sphere in Diagram 3B with the correct region of and the

volume element giving the correct order of integration:

(a) Limits: 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 3, 0 ≤ r ≤ z or:

ˆ 2π0

ˆ 30

ˆ z

0

Region of Integration: Cone (Diagram 3A) Sphere (Diagram 3B)

Order of Integration: dV = r dz dr dθ dV = r dr dz dθ

(b) Limits: 0 ≤ θ ≤ 2π, −3 ≤ z ≤ 3, 0 ≤ r ≤√

9− z2 or:

ˆ 2π0

ˆ 3−3

ˆ √9−z20



(c) Limits: 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 3, r ≤ z ≤ 3 or:

ˆ 2π0

ˆ 30

ˆ 3r



(d) Limits: 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 3, −√

9− r 2 ≤ z ≤√

9− r 2 or:

ˆ 2π0

ˆ 30

ˆ √9−r2−√9−r2



(Q15) Using your answers from (Q14), set up (but do not evaluate) the triple integral in cylindrical

coordinates that gives the volume of the sphere x2 + y 2 + z2 = 9, using the given order of

integration:

(a) dV = r dz dr dθ:

(b) dV = r dr dz dθ:


(Q16) Using your answers from (Q14), set up (but do not evaluate) the triple integral in cylindrical

coordinates that gives the mass of a solid region with density δ(x, y , z) = (x2+ y 2+ z)g/cm3,

bounded by the cone z =√x2 + y 2 and the plane z = 3, using the given order of integration:

(a) dV = r dz dr dθ:

(b) dV = r dr dz dθ:

(⊕ Q17) Set up and evaluate a triple integral in cylindrical coordinates that gives the volume of the

“napkin ring” formed by drilling a cylinder x2 + y 2 = R2 out of a sphere x2 + y 2 + z2 = P 2

(where R and P are constants with 0 ≤ R ≤ P ).

worksheet 3.4: triple integrals in cylindrical...

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