8/10/2019 math solution

1/3

Assignment 5 (MATH 215, Q1)

1. Evaluate the triple integral.

(a) E

xy dV, where E is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0),(0, 2, 0), and (0, 0, 3).

Solution. The plane containing the three points (1, 0, 0), (0, 2, 0), and (0, 0, 3) has anequation 6x+ 3y+ 2z= 6. Thus,

E

xydV =

10

22x0

(66x3y)/20

xy dz dy dx

=

10

22x0

1

2(6 6x 3y)xy dy dx

= 1

0 2(x4

+ 3x

3

3x2

+x) dx=

1

10 .

(b)

Ex dV, where E is bounded by the paraboloid x = 4y2 + 4z2 and the plane

x= 4.Solution. We have Q= {(y, z) :y2 +z2 1} and

E

xdV =

Q

44y2+4z2

x dx

dA=

Q

8 8(y2 +z2)2 dA

=

20

10

(8 8r4)rdrd= 2 10

(8r 8r5) dr= 163

.

2. (a) Find the volume of the region inside the cylinder x2 +y2 = 9, lying above thexy-plane, and below the plane z = y + 3.

Solution. We have Q= {(x, y) :x2 +y2 9} and

V =

E

dV =

Q

y+30

dz

dA=

Q

(y+ 3) dA

=

20

30

(r sin + 3) rdrd=

20

(9 sin + 27/2) d= 27.

(b) Find the volume of the region bounded by the paraboloids z = x2 +y2 andz = 36 3x2 3y2.

Solution. We have Q= {(x, y) :x2 +y2 9} and

V =

E

dV =

Q

363x23y2x2+y2

dz dA=

Q

(36 4x2 4y2) dA

=

20

30

(36 4r2)rdrd= 2 30

(36r 4r3) dr= 162.

1

8/10/2019 math solution

2/3

3. Use cylindrical coordinates in the following problems.(a) Evaluate

E

x2 +y2 dV, where E is the solid bounded by the paraboloid

z = 9 x2 y2 and the xy-plane.Solution. In cylindrical coordinates the region Eis described by

0 r 3, 0 2, and 0 z 9 r2.

Thus,

E

x2 +y2 dV =

20

30

9r20

r r dz dr d= 20

30

r2(9 r2) dr d

=

20

d

30

(9r2 r4) d= 3245

.

(b) Evaluate the integral E

x2 dV, whereEis the solid that lies within the cylinderx2 +y2 = 1, above the plane z = 0, and below the cone z2 = 4x2 + 4y2.

Solution. In cylindrical coordinates the region Eis described by

0 r 1, 0 2, and 0 z 2r.

Thus, E

x2 dV =

20

10

2r0

(r cos )2 r dz dr d

=

20

cos2 d

10

2r4 dr=2

5

.

4. Use spherical coordinates in the following problems.

(a) Evaluate

Ex e(x2+y2+z2)2 dV, whereEis the solid that lies between the spheres

x2 +y2 +z2 = 1 and x2 +y2 +z2 = 4 in the first octant

{(x,y,z) :x 0, y 0, z 0}.

Solution. In spherical coordinates the region Eis described by

1 2, 0 /2, 0 /2.

Thus,

E

x e(x2+y2+z2)2 dV =

/20

/20

21

( sin cos ) e4

2 sin ddd

=

/20

sin2 d

/20

cos d

21

3e4

d

=

16(e16 e).

2

8/10/2019 math solution

3/3

(b) Evaluate 33

9x29x2

9x2y20

z

x2 +y2 +z2 dz dy dx.

Solution

. The integral is equal to

Ez

x2

+y2

+z2

dV, where

E= {(,,) : 0 3, 0 2, 0 /2}.

Therefore, the integral is equal to

/20

20

30

( cos ) 2 sin ddd

=

/20

cos sin d

20

d

30

4 d=243

5 .

5. (a) Find the center of mass of the solidSbounded by the paraboloid z = 4x2 + 4y2

and the plane z = 1 ifShas constant density K.

Solution. In cylindrical coordinates the region Eis described by

0 r 1/2, 0 2, and 4r2 z 1

Thus, the mass of the solid is

M=

E

K dV =

20

1/20

14r2

Kr dz dr d=K

8 .

The moment about the xy-plane is

Mxy =

E

zK dV =

20

1/20

14r2

Kz r dz dr d=K

12 .

Similarly, the other two moments are Mxz = Myz = 0. We have Mxy/M = 2/3.Hence, the center of mass is (0, 0, 2/3).

(b) Find the mass of a ball given byx2 +y2 +z2 a2 if the density at any point isproportional to its distance from the z-axis.

Solution. In spherical coordinates the region Eis described by

0 a, 0 2, 0 .

The density is k sin , where k is a constant. Hence, the mass is

M=

E

k sin dV =

20

0

a0

(k sin )2 sin ddd=k2a4

4 .

3