Section 9.7/12.7: Triple Integrals in Cylindrical and Spherical Coordinates

Practice HW from Larson Textbook (not to hand in)Section 9.7: p. 606 # 1-35 odd

Section 12.7: p. 780 # 1-4 , Supplemental Exercises at end of notes # 1-8

Cylindrical Coordinates

Cylindrical coordinates extend polar coordinates to 3D space. In the cylindrical coordinate system, a point P in 3D space is represented by the ordered triple . Here, r represents the distance from the origin to the projection of the point P onto the x-y plane, is the angle in radians from the x axis to the projection of the point on the x-y plane, and z is the distance from the x-y plane to the point P.

As a review, the next page gives a review of the sine, cosine, and tangent functions at basic angle values and the sign of each in their respective quadrants.

1

x

y

z

r

Sine and Cosine of Basic Angle Values

Degrees Radians

0 0 0

30

45 1

60

90 0 1 undefined

180 -1 0 0

270 0 -1 undefined

360 1 0 0

Signs of Basic Trig Functions in Respective Quadrants

Quadrant

I + + +II - + -III - - +IV + - -

The following represent the conversion equations from cylindrical to rectangular coordinates and vice versa.

Conversion Formulas

To convert from cylindrical coordinates to rectangular form (x, y, z) and vise versa, we use the following conversion equations.

From cylindrical to rectangular form: , , z = z.

From rectangular to cylindrical form: , , and z = z

2

Example 1: Convert the points and from rectangular to cylindrical coordinates.

Solution:

3

█

4

Example 2: Convert the point from cylindrical to rectangular coordinates.

Solution:

█

Graphing in Cylindrical Coordinates

Cylindrical coordinates are good for graphing surfaces of revolution where the z axis is the axis of symmetry. One method for graphing a cylindrical equation is to convert the equation and graph the resulting 3D surface.

5

Example 3: Identify and make a rough sketch of the equation .

Solution:

█

Example 4: Identify and make a rough sketch of the equation .

Solution:

█

6

x

y

z

x

y

z

Spherical Coordinates

Spherical coordinates represents points from a spherical “global” perspective. They are good for graphing surfaces in space that have a point or center of symmetry.

Points in spherical coordinates are represented by the ordered triple

where is the distance from the point to the origin O, , where is the angle in radians from the x axis to the projection of the point on the x-y plane (same as cylindrical coordinates), and is the angle between the positive z axis and the line segment joining the origin and the point P . Note .

7

x

y

z

Conversion Formulas

To convert from spherical coordinates to rectangular form (x, y, z) and vise versa, we use the following conversion equations.

From spherical to rectangular form: , ,

From rectangular to spherical form: , , and

Example 5: Convert the points (1, 1, 1) and from rectangular to spherical coordinates.

Solution:

8

█

Example 6: Convert the point from rectangular to spherical coordinates.

9

Solution:

█

Example 7: Convert the equation to rectangular coordinates.

Solution:

█

Example 8: Convert the equation to rectangular coordinates.

10

Solution: For this problem, we use the equation . If we take the

cosine of both sides of the this equation, this is equivalent to the equation

Setting gives

.

Since , this gives

or

Hence, is the equation in rectangular coordinates. Doing some algebra will help us see what type of graph this gives.

Squaring both sides gives

The graph of is a cone shape half whose two parts be found by graphing the two equations . The graph of the top part, , is displayed as follows on the next page.

(continued on next page)

11

█

Example 9: Convert the equation to cylindrical coordinates and spherical coordinates.

Solution: For cylindrical coordinates, we know that . Hence, we have or

For spherical coordinates, we let , , and to obtain

We solve for using the following steps:

█Triple Integrals in Cylindrical Coordinates

Suppose we are given a continuous function of three variables expressed over a solid region E in 3D where we use the cylindrical coordinate system.

12

Then

13

x

y

z

E

Example 10: Use cylindrical coordinates to evaluate , where E is the

solid in the first octant that lies beneath the paraboloid .

Solution:

14

█

15

Example 11: Use cylindrical coordinates to find the volume of the solid that lies both within the cylinder and the sphere .

Solution: Using Maple, we can produce the following graph that represents this solid:

In this graph, the shaft of the solid is represented by the cylinder equation . It is capped on the top and bottom by the sphere . Solving for z, the upper and bottom portions of the sphere can be represented by the equations .Thus, z ranges from to . Since in cylindrical coordinates, these limits become to .When this surface is projected onto the x-y plane, it is represented by the circle . The graph is

(Continued on next page)

16

This is a circle of radius 2. Thus, in cylindrical coordinates, this circle can be represented from r = 0 to r = 2 and from to . Thus, the volume can be represented by the following integral:

We evaluate this integral as follows:

Thus, the volume is .

█

Triple Integrals in Spherical Coordinates

Suppose we have a continuous function defined on a bounded solid region E.

17

x y

z

Then

18

Example 12: Use spherical coordinates to evaluate , where Q is

enclosed by the sphere in the first octant.

Solution:

19

█

20

Example 13: Convert from rectangular to

spherical coordinates and evaluate.

Solution: Using the identities and , the integrand becomes

The limits with respect to z range from z = 0 to . Note that is a hemisphere and is the upper half of the sphere .

The limits with respect to y range from y = 0 to , which is the semicircle located on the positive part of the y axis on the x-y plane of the circle as x ranges from to . Hence, the region described by these limits is given bythe following graph

Thus, we can see that ranges from to , ranges from to and

ranges from to . Using these results, the integral can be evaluated in polar coordinates as follows:

(continued on next page)

21

█

22

Supplemental Exercises

1. Use cylindrical coordinates to evaluate , where Q is the region that lies

inside the cylinder and between the planes and . (Answer )

2. Use cylindrical coordinates to evaluate , where Q is the region enclosed by

the paraboloid , cylinder , and the x-y plane. (Answer )

3. Find the volume of the region Q bounded by the paraboloids and . (Answer )

4. Use spherical coordinates to evaluate , where Q is the unit ball

. (Answer )

5. Use spherical coordinates to evaluate , where Q lies between the spheres

and in the first octant. (Answer )

6. Use spherical coordinates to evaluate , where Q is bounded by the xy-plane

and the hemispheres and . (Answer )

7. Evaluate the integral by changing to cylindrical

coordinates. (Answer 0)

8. Evaluate the integral by changing to cylindrical

coordinates. (Answer )

23