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Additional Review Questions for Old Test 3 - Calculus II | MTH 252, Exams of Calculus

Material Type: Exam; Class: Calculus II; Subject: Math; University: Portland Community College; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Additional Review Questions for MTH 252 Test 3
All integrals may be and should be evaluated on your calculator unless otherwise stated.
1. Use the washer method the find the volume of the solid that results spinning the region
outlined in Figure 1 about the line 4
โˆ’
=
y. To earn full credit Figure 1 must be annotated
with all of the labels discussed and illustrated in class.
6. Use the shell method to find the volume of the solid created by rotating about the line
2โˆ’=y the region enclosed by the parabola 2
xy = and the line 4
=
y. To earn full credit
Figure 2 must be annotated with all of the labels discussed and illustrated in class.
Figure 1
Figure 2
pf3
pf4
pf5

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Additional Review Questions for MTH 252 Test 3

All integrals may be and should be evaluated on your calculator unless otherwise stated.

  1. Use the washer method the find the volume of the solid that results spinning the region

outlined in Figure 1 about the line y =โˆ’ 4. To earn full credit Figure 1 must be annotated

with all of the labels discussed and illustrated in class.

  1. Use the shell method to find the volume of the solid created by rotating about the line

y =โˆ’ 2 the region enclosed by the parabola

2 y = x and the line y = 4. To earn full credit

Figure 2 must be annotated with all of the labels discussed and illustrated in class.

Figure 1

Figure 2

  1. Estimate

2 2 1

x 1 dx โˆ’

using 6 trapezoids over intervals of equal width.

  1. Estimate

5 3 1

x dx

using Simpsonโ€™s Rule with 2 n = 10 intervals of equal width.

  1. Sketch onto figure 3 a complete graph of the parametric equations indicated in the Figure

caption; then use an appropriate integral to determine the arc-length along the entire curve.

6. Sketch onto Figure 4 the function ( )

2

3 cos 2

y x = โˆ’ y y โˆ’. Pay special attention to where you

plot the points where the curve goes off the grid and at each point the curve changes

direction (left to right).

Figure 3

2cos 2 4, 5sin 1 2

x t y t

Figure 4

2

10 cos 1. 2

y x = + y y โˆ’

Additional Review Questions for MTH 252 Test 3 โ€“ Answers

1. ( ) ( ) [ ]

2 (^2 2 )

4

b

a

V ฯ€ R x r x dx ฯ€ x x dx ฯ€

โˆ’

4

0

d

c

V y h y dy y y dy

x

ฮ” = = Define ( )

2 f x = x + 1. Then

(^2 )

1 1 6

x dx T

f f f f f f f

โˆ’

x

ฮ” = = Define ( )

3 f x = x. Then

5 3 10 1

1 4 1.4 2 1.8 4 2.2 2 2.6 4 3 2 3.4 4 3.8 2 4.2 4 4.6 5 3 156

x dx S

f f f f f f f f f f f

โ‰ˆ

= โŽกโŽฃ + + + + + + + + + + โŽคโŽฆโ‹…

=

Figure 3

2 2

2 (^3 2 ) 4sin 2 cos 2 2

b

a

dx dy L dt dt dt

t t dt

ฯ€

ฯ€

โŽ โŽ^ โŽ โŽ 

Figure 4

1

2

1 2 1 2

2

5 /

5

5 /

2 5

2

2

2

sin sin

5 lim

sin 5 lim^1

lim cos 5

lim cos cos 5 5

cos 0 cos 5 5

t t t t t t t

u x dx dx x x (^) du x dx

x x du dx

du x u

x u

x

u

t

ฯ€

ฯ€

ฯ€

ฯ€

โˆ’

โˆž โˆ’

โ†’ โˆž

โ†’ โˆž

โ†’ โˆž

โ†’ โˆž

= =^ โ‡’^ =
โŽ โŽ^ โŽ^ โŽ  โŽ โŽ 

t โ‡’ u = 5 ฯ€/ t

7. From Hooke's law we know that F ( x ) = kx where x is the displacement of the spring from its

natural length. Since F ( ) 2 = 8 = 2 k , k = 4. Hence the work done while stretching the

spring from its natural length to 4 inches beyond its natural length is given by:

4

0

W xdx

32 in-lb of work is done when stretching the spring from a length of 20 in to a length of 24 in.

  1. a.

b.

5 5

4 4 4 4 4 3 3

lim lim 2 6 2 6 2 6

t

t t t

dx dx dx

x x x

โˆ’ (^) โ†’ โˆ’^ โˆ’ โ†’+

2 6 4 4 4 14 3 3

3 2 6

3 14

3 3 3

lim lim 2 6 2 2

lim (^2) 2 3 4 14

lim (^6 2 6 )

t t

t t

t

t

t

dx du u x u x du^ dx

du u dx

x u

x t u t

t

โˆ’ โˆ’

โˆ’

โˆ’

โˆ’ โˆ’

โ†’ โˆ’^ โ†’ โˆ’

โˆ’ โˆ’

โ†’ โˆ’

โ†’

= โŽœ^ โ‹… โŽŸ
= โŽœ^ โˆ’ โŽœ^ โˆ’ โŽŸโŽŸ
โŽœ โŽœ^ โŽŸโŽŸ
โŽ โŽ^ โŽ โŽ