Math 2263 Quiz 10 - University of Minnesota

Math 2263 Quiz 10

26 April, 2012

Name:

1. Evaluate S z dS, where S is the part of the plane 2x + 2y + z = 4 that lies in the first octant.

Answer: The x-, y-, and z-intercepts of the given plane are 2, 2, and 4. Thus S is the triangular region with vertices (2, 0, 0), (0, 2, 0), and (0, 0, 4). The We choose x and y to be parameters, the domain for the parameters is the triangular region with vertices (0, 0), (2, 0), and (0, 2), i.e.,

D = (x, y) R2 : 0 x 2, 0 y 2 - x .

The plane is z = 4 - 2x - 2y, and hence z z = = -2. x y

So we have

z dS = (4 - 2x - 2y) (-2)2 + (-2)2 + 12 dA

S

D

2 2-x

=

6(2 - x - y) dy dx

00

2

y=2-x

= 12y - 6xy - 3y2

dx

0

y=0

2

= 3x2 - 12x + 12 dx

0

x=2

= x3 - 6x2 + 12x

x=0

= 8.

1

2. Find S F ? dS, where F = xy i + yz j + zx k, S is part of the paraboloid z = 4 - x2 - y2 that lies above the square 0 x 1, 0 y 1, and has upward orientation.

Answer: We use Equation 10 in Section 16.7.

z

z

= -2x, = -2y.

x

y

So we have

F ? dS =

-(-2x)xy + -(-2y)y(4 - x2 - y2) + (4 - x2 - y2)x dA

S

[0,1]?[0,1]

11

=

2x2y + 2y2(4 - x2 - y2) + (4 - x2 - y2)x dy dx

00

11

=

-2y4 - (2x2 + x - 8)y2 + 2x2y + (-x3 + 4x) dy dx

00

= 1 - 2 y5 - 2x2 + x - 8 y3 + x2y2 + (-x3 + 4x)y y=1 dx

0

5

3

y=0

=

1

-x3

+

1 x2

+

11 x

+

34

dx

0

3

3 15

=

1 -

x4

+

1 x3

+

11 x2 +

34 x

1

49

6

15 0

713 =.

180

2

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download