Test 1 Review Answers

Math 214 Spring 2001

 

1) Page 471 #2

 

The integral is = 16 1/5 = 81/5

 

2) Page 471 #16

 

Vertical rectangles:

 

Horizontal:

 

Horizontal is obviously easier. Compute to get area = 4/3

 

3) Page 471 #22

 

a) Discs: = 8p

 

 

b) Shell: Note: This is the same integral as part (a) if you

set (a) up using shells.

 

 

c) Discs:

 

 

d) Disc:

 

 

Shell:

 

Discs were easier here.

 

 

4) Page 472 #41

 

Draw the picture so the bottom of the well is centered about the origin, then

all the numbers are positive. For a representative disc, or cylinder:

 

where r = 1/3

 

the distance the disc must be lifted is 175 y feet. So,

 

 

 

 

5) Page 472 #46

 

Notice that the area under the curve has the units of (pounds)(feet) = ft-lbs which

are the units for work. So, find the area under the curve:

 

Between 0 and 9 the curve is the straight line y = -2/9 x + 6

 

Between 9 and 12, the curve is the line y = -4/3 x + 16

 

Do the integration to get the total area : 45 + 6 = 51

 

 

6) Page 472 #48

 

 

The centroid is just the center of mass for an object of constant density. Recall

that r always cancels in the division when it is constant.

 

Need mass = = 32/3

 

 

Moment about the x-axis: = 544/15

 

 

Moment about the y-axis: = 32/3

 

This results in: I hope.