Math 1B
Exam 1 Review
The first exam will be on Friday, March 4th. It will cover sections 5.3, 5.4, 5.5 and chapter 6. The exam
will consist of questions worth various values for a total of 125 points. You are allowed to use one 3” by
5” notecard (no specific examples, no unit circle, no table of integrals) and a scientific calculator (no
graphing calculators). Prepare well and good luck!
You will need to be able to:
1.
2.
3.
4.
5.
6.
7.
8.
Apply the Fundamental Theorem of Calculus Part 1. (5.3)
Evaluate definite integrals using the Fundamental Theorem of Calculus Part 2. (5.3)
Evaluate indefinite integrals. (5.4)
Use substitution to evaluate definite and indefinite integrals. (5.5)
Find areas between curves. (6.1)
Find volumes of solids of revolution using disks, washers, and shells. (6.2, 6.3)
Solve work problems. (6.4)
Find the average value of a function on a given interval. (6.5)
Check out the Chapter 5 Review Exercises, page 417, # 9-19 odd, 23-35 odd, 43-46
Check out the Chapter 6 Review Exercises, pages 457-458, # 1-10, 12, 14, 15, 16, 27, 28, 30
Additional Problems:
1.
Find the derivative of each function.
a) g(x) =
2.
∫
x2
0
3.
€
4.
5.
∫
sin x
3
1
dt
1+ t 2
Evaluate each integral.
€
€
b) h(x) =
ln( t + 1) dt
a)
∫ xe
e)
∫x
−x 2
€
b) ∫ ( 3t − 2) dt
4
dx
1− xdx
€
f)
1
ex
∫ 1+ e
0
2x
dx
€
sin z
dz
z
d)
∫
sin θ
dθ
cos2 θ €
h)
∫
π2
c)
∫
g)
∫
0
π
3
0
cos(ln x)
dx
x
π
4
0
tan x sec 2 xdx
Find the area of the region enclosed by the curves y = x 2 and y = 2x − x 2 .
€
€
€
Find the volume of the solid obtained when the region enclosed by y = x 2 and y = 2x − x 2 is
revolved about the x-axis. You can use any method, but identify the type of object made by your
€
€
typical slice (disk, washer, or shell).
Let R be the region enclosed by the curves y = x 2 and y =€ x .
€
a) Set up the integral necessary to find the volume of the solid obtained by revolving R about
the x-axis. You do not need to integrate, but identify the type of object made by your typical
€
slice (disk, washer, or shell). €
b) Set up the integral necessary to find the volume of the solid obtained by revolving R about
the y-axis. You do not need to integrate, but identify the type of object made by your typical
slice (disk, washer, or shell).
6.
€
7.
1
, y = 0 , x = 1, and
x
x = 4 is revolved about the x-axis. You can use any method, but identify the type of object made
by your typical slice (disk, washer, or shell).
€
€
€
Find the volume of the solid obtained when the region enclosed by y = x and y = x is revolved
about the specified line. You can use any method, but identify the type of object made by your
typical slice (disk, washer, or shell).
Find the volume of the solid obtained when the region enclosed by y =
€
a) x = 2
b) y = −1
8.
€
€
9.
€
A 500 lb load of coal is pulled up a 100 ft deep mineshaft using a chain weighing 0.5 lb/ft. Find
the work required to lift the chain and the coal out of the mineshaft.
A swimming pool is built in the shape of a rectangular box 10 ft deep, 15 ft wide, and 20 ft long.
If the pool is filled to 1 ft below the top, how much work is required to pump all the water into a
drain at the top edge of the pool? (Use the fact that water weighs 62.5 lb/ft3.)
10. A tank is full of water. Find the work required to pump the water
out of the spout. (Use 9.8 m/s2 for g. Use 1000 kg/m3 as the density
of water. Assume r = 3 m and h = 1 m.)
11. Suppose that a conical tank (with the cone point on the bottom) has height 10 ft, the radius of the
base is 7 ft, and it is half filled with water. Find the amount of work necessary to move all of the
water out of the top of the tank. (Use the fact that water weighs 62.5 lb/ft3.)
12. If the temperature in a controlled experiment is given by the function T(t) = t 2 + 1, where t is the
time in hours and T is the temperature in degrees Celsius.
a) Find the average temperature over the first 3 hours.
€
b) Find the time (between 0 and 3 hours) where the temperature
equals the average temperature.
c) Graph the temperature function between t = 0 and t = 3. Shade the region below the curve
and draw a rectangle with the same base (3 units) whose area would equal the area under the
temperature function.
€
€
Answers to Additional Problems
1.
2.
€
a) g"(x) = 2x ln( x 2 + 1)
b) h "(x) =
1 2
a) − e−x + C
2
b)
1
(3t − 2)5 + C
15
5
3
2
2
e) (1− x ) 2 − (1− x ) 2 + C
5
3
2
h)
3
€
d) sin(ln x) + C
€
€
3.
g) 1
€
1
3
π
3
5.
a) If you use washers:
€
∫ π $%&( x ) − ( x ) '()dx
V = ∫ 2πy ( y − y ) dy
V=
1
2
2 2
0
1
or if you use shells:
2
0
€
b) If you use shells:
1
V=
∫ 2πx (
V=
∫ π $&%( y ) − ( y ) ')(dy
0
)
x − x 2 dx
€
6.
or if you use washers:
€
3π
4
€
8π
15
7.
a)
8.
52,500 ft-lb
€
€9.
b)
π
2
€
928,125 ft-lb
10. ≈ 4,433,416 J
11. ≈ 25,055 ft-lb
12. a) Tave = 40
b) t = 3 ≈ 1.732 hours
€
c) 4
f) tan−1 (e) −
€
€
4.
€
cos x
1+ sin 2 x
1
0
2
2 2
π
4