296 STEP 4. Review the Knowledge You Need to Score High
For Problems 16 through 19, find the
volume of the solid obtained by revolving
the region as described below.
(See Figure 12.7-2.)
R 1
C(0,8)
A(2,0)
R 2
y
x
B(2,8)
y = x^3
0
Figure 12.7-2
- R 1 about thex-axis.
- R 2 about they-axis.
- R 1 about the line
←→
BC.
- R 2 about the line
←→
AB.
- The functionf(x) is continuous on [0, 12]
and the selected values off(x) are shown in
the table.
x 0 2 4 6 8 10 12
f(x) 1 2.24 3 3.61 4.12 4.58 5
Find the approximate area under the curve
off from 0 to 12 using three midpoint
rectangles.
- Find the area bounded by the curve defined
byx=2 costandy=3 sintfromt=0to
t=π. - Find the length of the arc ofr=sin^2
(
θ
2
)
fromθ=0toθ=π.
- Find the area of the surface formed when
the curve defined byx=etsintand
y=etcostfromt=0tot=
π
2
is revolved
about thex-axis.
- Find the area bounded byr= 2 +2 sinθ.
- The acceleration vector for an object is
〈−et, et〉. Find the position of the object at
t=1 if the initial velocity isv 0 =
〈
3, 1
〉
and
the initial position of the object is at the
origin.
12.8 Cumulative Review Problems
(Calculator) indicates that calculators are
permitted.
- If
∫a
−a
ex
1
dx=k, find
∫a
0
ex
2
dxin terms
ofk.
- A man wishes to pull a log over a 9 foot
high garden wall as shown in
Figure 12.8-1. He is pulling at a rate of
2 ft/sec. At what rate is the angle between
the rope and the ground changing when
there are 15 feet of rope between the top of
the wall and the log?
θ
9 ft
wall
rope
rope
log
Figure 12.8-1
- (Calculator) Find a point on the parabola
y=
1
2
x^2 that is closest to the point (4, 1).