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1 Section 9.7/12.8: Triple Integrals in Cylindrical and Spherical

Coordinates

Practice HW from Stewart Textbook (not to hand in)

Section 9.7: p. 689 # 3-23 odd

Section 12.8: p. 887 # 1-11 odd, 13a, 17-21 odd, 23a, 31, 33

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 ),,(zr. 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. x y z r ),,(zrP 2

Sine and Cosine of Basic Angle Values

Degrees Radians cos sin

T T cossintan

0 0 10cos 00sin 0

30
6 23
21
33
45
4 22
22
1 60
3 21
23
3 90
2

0 1 undefined

180 -1 0 0

270
23

0 -1 undefined

360 2 1 0 0

Signs of Basic Trig Functions in Respective Quadrants

Quadrant

cos sin T T cossintan

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 ),,(zr to rectangular form (x, y, z) and vise versa, we use the following conversion equations.

From polar to rectangular form:

cosrx, sinry, z = z.

From rectangular to polar form:

222
yxr , xy tan , and z = z 3 Example 1: Convert the points )3,2,2( and )1,3,3( from rectangular to cylindrical coordinates.

Solution:

4

Example 2: Convert the point )1,4,3(

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 2 rz.

Solution:

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

Solution:

x y z x y z 6

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 OP joining the origin and the point P ),,( . Note 0. x y ),,(P z 7

Conversion Formulas

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

From to rectangular form:

cos sinx, sin siny, cosz

From rectangular to polar form:

2222
zyx , xy tan , and )arccos()arccos( 222
z zyxz Example 5: Convert the points (1, 1, 1) and )22,3,3( from rectangular to spherical coordinates.

Solution:

8

Example 6: Convert the point ),4,9(

S from rectangular to spherical coordinates.

Solution:

Example 7:

Convert the equation sec2 to rectangular coordinates.

Solution:

9

Example 8: Convert the equation

3 to rectangular coordinates. Solution: For this problem, we use the equation )arccos( 222
zyxz . If we take the cosine of both sides of the this equation, this is equivalent to the equation 222
cos zyxz

Setting

3 gives 222
3cos zyxz S Since 21
3cos S , this gives 222
21
zyxz or zzyx2 222

Hence, zzyx2

222
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 03

222
zyx is a cone shape half whose two parts be found by graphing the two equations zzyx2 222
. The graph of the top part, zzyx2 222
, is displayed as follows on the next page. (continued on next page)

034)2(

22222222222

zyxzzyxzzyx 10

Example 9:

Convert the equation zyx

22
to cylindrical coordinates and spherical coordinates. Solution: For cylindrical coordinates, we know that 222
yxr. Hence, we have zr 2 or zr

For spherical coordinates, we let

cos sinx, sin siny, and cosz to obtain cos)sin sin()cos sin( 22

We solve for

using the following steps: 22

22222222222222222

sincos,0 solve) and zero toequalfactor each (Set 0cossin,0 )(Factor 0)cossin( )1sincosidentity (Use 0cos)1( sin )sin(Factor cos)sin(cossin terms)(Square cossin sincos sin

11

Triple Integrals in Cylindrical Coordinates

Suppose we are given a continuous function of three variables ),,(zrf expressed over a solid region E in 3D where we use the cylindrical coordinate system. Then 2 12 12 1 gr grrhz rhzE ddrdzrzrfdVzrf x y z 2 1 2 rhz 1 rhz 1 gr 2 gr E 2 12 12 1

E of Volume

gr grrhz rhzE ddrdzrdV 12 Example 10: Use cylindrical coordinates to evaluate E dVxyx )( 23
, where E is the solid in the first octant that lies beneath the paraboloid 22
1yxz.

Solution:

13 Example 11: Use cylindrical coordinates to find the volume of the solid that lies both within the cylinder 4 22
yx and the sphere 9 222
zyx. 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 4 22
yx. It is capped on the top and bottom by the sphere 9 222
zyx. Solving for z, the upper and bottom portions of the sphere can be represented by the equations 22
9yxz.

Thus, z ranges from

22

9yxz to

22

9yxz. Since

222
ryx in cylindrical coordinates, these limits become 2

9rz to

2

9rz.When this surface

is projected onto the x-y plane, it is represented by the circle 4 22
yx. The graph is (Continued on next page) 14 This is a circle of radius 2. Thus, in cylindrical coordinates, this circle can be represented from r = 0 to r = 2 and from

0T to 2. Thus, the volume can be represented by the

following integral: 2 02 09 9)( 2 22
12 12 1 r rrz rzgr grrhz rhzE ddrdzrddrdzrdVVolume

We evaluate this integral as follows:

532036 02)5310(18 ]5310[18 )55)5( and 27 )9( (Note ]5310[18 ])9(32)5(32 [ ])09(32)29( 32[ )-

9 let subdu -u (Use )9( 32 92 )9()9(

2 02 02 02 2 0 22
2 02 0 22
2 02 0222
02 09 92
02 02 02 09 9 23
232
3 232
3 232
32
22
2 dddru drddrrr ddrrrr rddrrz ddrdzr r rr rr rrz rzr rr rrz rz

Thus, the volume is 532036

15

Triple Integrals in Spherical Coordinates

Suppose we have a continuous function ),,(f defined on a bounded solid region E. Then 2 12 12 1 ),(2 sin ),,( ),,( h hE dddfdVf 2 12 12 1 ),(2 sin E of Volume h hE ddddV x y ),,(P z 16

Example 12: Use spherical coordinates to evaluate

Ezyx dVe 222
, where E is enclosed by the sphere 9 222
zyx in the first octant.

Solution:

17

Example 13: Convert

2 24
016 0 22
222
xyx ddydzyxfrom rectangular to spherical coordinates and evaluate. Solution: Using the identities cossinx and sinsiny, the integrand becomes sin)1(sin )sin(cossin sinsincossin

22222222222222

yx

The limits with respect to z range from z = 0 to

22

16yxz. Note that

22

16yxz is a hemisphere and is the upper half of the sphere 16

222
zyx.

The limits with respect to y range from y = 0 to

2

4xy, which is the semicircle

located on the positive part of the y axis on the x-y plane of the circle 4 22
yx as x ranges from

2x to 2x. Hence, the region described by these limits is given by

the following graph

Thus, we can see that

ranges from 0 to 4, ranges from 0I to 2 and ranges from 0T to . Using these results, the integral can be evaluated in polar coordinates as follows: (continued on next page) 18quotesdbs_dbs20.pdfusesText_26