Z ENGR 233 Winter 2018 webwork #7 PDF

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Feras Badr Bahlas Assignment Assignment 7 due 03/20/2018 at 11:58pm EDT

Z ENGR 233 Winter 2018

• G • D • H

1. (1 point) Match each double integral in polar with the graph of the region of integration.

? 1. ? 2. ? 3.

Z 2π Z 5

Z0 Z0

0

? 4. ? 5. ? 6. ? 7. ? 8.

Z 4/√2 Z

f (r, θ)r dr dθ

0 4 Z 3π/4

f (r, θ)r dθ dr

f (r, θ)r dr d θ

3π/2 0 Z 5 Z 7π/4

Z BZ D

f (r, θ)r dθ dr

3π/4

Z 2π Z 4 3π/4

0

−π/4

4

C

A

f (r, θ)r dr dθ

Z0 3π/20Z 4

Z 3π/4 Z 5

xy dx dy

Instructions: Please enter the integrand in the first answer box, typing theta for θ. Depending on the order of integration you choose, enter dr and dtheta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration and evaluate the integral to find the volume.

f (r, θ)r dθ dr

−π/2

Z 2π Z 5

16−y2

y

0

4 3π/4

4Z

4

2. (1 point) Convert the integral below to polar coordinates and evaluate √ the integral.

A= B= C= D=

f (r, θ)r dr dθ f (r, θ)r dr dθ

Volume = Correct Answers: • rˆ2*cos(theta)*sin(theta)*r; dr; dtheta; 0; pi/4; 0; 4 • 16 2 −y2

A

B

3. (1 point) Consider the solid under the graph of z = e−x above the disk x2 + y 2 ≤ a2 , where a > 0.

C

(a) Set up the integral to find the volume of the solid.

D

E

Instructions: Please enter the integrand in the first answer box, typing theta for θ. Depending on the order of integration you choose, enter dr and dtheta in either order into the second and third answer boxes with only one dr or dtheta in each box. Then, enter the limits of integration.

F

Z BZ D A

G Correct Answers: • C • F • E • B • A

C

A= B= C= D=

H

(b) Evaluate the integral and find the volume. Your answer will be in terms of a. Volume V = 1

(c) What does the volume approach as a → ∞? lim V =

• • • • •

a→∞

Correct Answers: • eˆ(-rˆ2)*r; dr; dtheta; 0; 2*pi; 0; a • pi*[1-eˆ(-aˆ2)] • 3.14159

1 4 f(x,y) y x

7. (1 point) Find the volume of the wedge-shaped region (Figure 1) contained in the cylinder x2 + y 2 = 64 and bounded above by the plane z = x and below by the xy -plane.

4. (1 point) Find the volume of the finite region between the graph of f (x, y) = 4 − x2 − y 2 and the xy plane. volume = Correct Answers: • pi*16/2

5. (1 point) A disk of radius 7 cm has density 10 g/cm2 at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. mass = (Include units.) Correct Answers: • 513.127 g

6. (1 point) For each of the following, set up the integral of an arbitrary function f (x, y) over the region in whichever of rectangular or polar coordinates is most appropriate. (Use t for θ in your expressions.) (a) The region

V= Correct Answers: • 341.333

With a = ,b= c= , and d = R R integral = ab cd (b) The region

With a = ,b= c= , and d = R R integral = ab cd

, , d

d

d

d

4

8. (1 point) p RR Evaluate D x2 + y 2 dA, where D is the domain in Figure

, ,

Correct Answers: • 0 • 0.785398 • 0 • 2 • r*f(r*cos(t),r*sin(t)) • r • t • 2 • 3

F : x2 + y 2 = 256 G : (x − 8)2 + y 2 = 64 R f = 16 Rg = 8 2

RR p D

x2 + y 2 dA =

Correct Answers: • 6758.2

9. (1 point) Sketch the region of integration and evaluate by changing to polar coordinates: Z 8Z f (x) 4

0

√ For f (x) = 8x − x2 Answer:

1 p

x2 + y 2

dy dx

Correct Answers: • 2.13136

10. (1 point) For the region R below, write iterated integral in polar coordinates.

RR

R

f dA as an

(Click on graph to enlarge)

Correct Answers: • (3+5)*pi*16/8

With a = RR

R

f dA =

,b=

R bR d a c

,c=

, and d =

f dA, where dA =

, d

13. (1 point)  8UseGreen’s Theorem to evaluate the line integral of F = x , 3x around the boundary of the parallelogram in the following figure (note the orientation).

d

Note: Use t for θ in your expressions. Correct Answers: • • • • • • •

pi/2 3*pi/2 2 3 r r t R

11. (1 point) Calculate C ((3x +8y )~i+ (9x +6y) ~j)·d~r where C is the circular path with center (a, b) and radius m, oriented counterclockwise. Use Green’s Theorem. R ~ ~ C ((3x + 8y ) i + (9x + 6y) j) · d~r = Correct Answers:

With x0 = 5 and y 0 = 5. R 8 C x dx + 3x dy =

• (9-8)*pi*mˆ2

12. (1 point)

Correct Answers:

Suppose

• -75

~ F(x, y) = (2x − 3y)~i + 5x~j and C is the counter-clockwise oriented sector of a circle centered at the origin with radius 4 and central angle π/4. Use Green’s theorem to calculate the circulation of ~F around C.

14. (1 point) R Evaluate I = C (sin x + 2y)dx + (8x + y)dy for the nonclosed path ABCD in the figure.

Circulation = 3

Use Green’s Theorem to determine the circulation of F around C1 , assuming that F(x, y) = M (x, y)i + N (x, y) j and Nx − My = 9 on the shaded region.

A = (0, 0) , I=

B = (2, 2),

C = (2, 4) ,

D = (0, 6)

Correct Answers: • 66

15. (1 point) Referring to Figure 11, suppose that I

C2

F · ds = 8π,

I

C3

R

C1 F · ds =

Correct Answers:

F · ds = 9π

• 703.717

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