Fall 2010 Math 152
4
Night Before Drill for Exam I
1. Find the volume of the solid formed by rotating the region bounded by y = ln x, y =
x, y = 0, and y = 1 about the y-axis.
courtesy: David J. Manuel
2. Find the volume of the solid formed by rotating the region bounded by y = 2 sin x, y =
π
0, and x = about the y-axis.
2
(covering 6.4-8.2)
1
1. Find the derivative of F (x) =
ˆ
0
2. Compute
3. Compute
2
ˆ
1
xp
2 + t3 dt
4. The base of a solid is the ellipse x2 +4y 2 = 36.
Cross-sections perpendicular to the y-axis are
equilateral triangles. Find the volume of the
solid.
4
dx
1 + x2
0
6
0
|x − 2| dx
5
Section 6.5
1. Evaluate
ˆ 2
1
+ xe−x
x ln x
2. Evaluate
ˆ √
3. Compute
ˆ
3
3. Set up, but do not evaluate, an integral to find
the volume of the solid formed by rotating the
region bounded by x = 4−y 2 and x = 8−2y 2.
about the line y = 2.
Section 6.4
ˆ
Section 7.2-7.3
1. A force of 60 Newtons is required to hold a
spring that has been stretched from its natural length of 10cm to a length of 15cm. How
much work (in Joules) is required to stretch
the spring from its natural length to a length
of 20cm?
dx
x
x3
−√
1 − x2
1 − x2
Section 7.4
dx
π/6
2. The tank below is 6 meters long and the semicircular ends have a radius of 2 meters. If the
tank is full of water, find the work required
to pump all the water out of the tank.
tan(2x) dx
0
Section 7.1
1. Find the area of the region bounded by y =
x4 − x2 and y = 1 − x2 .
2. Find the
√ area of the region
√ bounded by y =
0, y = x + 1 and y = 3 − x.
3. A 50-foot cable with density 3 lbs/foot is used
to lift a 200-lb miner out from a cavern. Find
the work done.
3. Find the area of the region bounded by y =
ex , y = e3x , and x = 1.
1
6
Section 7.5
1. √
Find the average value of the function f (x) =
1 + 2x on the interval [1, 4].
2. On a certain day, the temperature t hours
past 10am is given by T (t) = 80 +
π
t . What was the average temper10 sin
12
ature between 2pm and 6pm?
7
Section 8.1
1. Compute
ˆ
e
x ln x dx
1
2. Evaluate
ˆ
x2 sin x dx.
3. Evaluate
ˆ
sin−1 x dx.
8
Section 8.2
1. Evaluate
ˆ
sin3 x cos11 x dx
2. Evaluate
ˆ
8 sec3 θ tan3 θ dθ.
3. Compute
ˆ
π
sin4 x dx.
0
2