Test 1 Review Answers

**Math 214 Spring 2001**

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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.