Unit 4 Integration Quiz Review (Ch. 4 in your textbook for more examples/practice)
Topics:
 4.1 Antiderivatives, indefinite integration (+C)
o Initial condition (to find C)
 4.2 Integral as the area under a curve
o Geometrically
o Riemann sums – left, right, midpoint
 4.3 Definite integrals
o Limit sum definition and process of evaluating integrals
o Properties
o 4.4 Fundamental Theorem of Calculus (Parts 1 and 2)  Use Versatiles
Activity as a study tool for Part 2 of FTC
o Average Value and Mean Value Theorem
 4.5 Integration by u-substitution and change of limits  Use Problem Set 19 and
Turvy Integration as study tools for this topic
 4.6 Numerical Integration
o Trapezoidal Rule
o Simpson’s Rule
Practice Problems:
4
1. Evaluate the definite integral
 (2x
2
 x  3)dx by the limit sum definition of
2
integrals.

2. A car going 80 ft/sec (about 50 mi/hr) brakes to a stop in 8 seconds. Its velocity is recorded
every 2 seconds and is given in the following table.
t (seconds)
v(t) (ft/sec)
0
80
2
52
4
28
6
10
8
0
A. Use the trapezoidal rule and Simpson’s rule to estimate the distance traveled by the car
during the 8 seconds. (n = 4)
B. Estimate the distance using a left-hand Riemann sum, right-hand Riemann sum, and
midpoint Riemann sum. For each, state if the value is an over or underestimate. (Think: is
the graph increasing or decreasing?)
1
3. If
6
6
2
2
  4 f ( x)  3 dx  24 , then find  f ( x)dx .
4. The average value of y  v(x) equals 4 for 1  x  6 and equals 5 for 6  x  8 . What is the
average value of v(x) for 1  x  8 ?
5. Without doing any calculations, find the values of

2
A.
 sin x dx
B.
 x
113
dx

2
6. For the even function  graphed in the figure:
A.
B.
Suppose you know
Suppose you know
2
2
0
2
 f ( x)dx . What is an expression for  f ( x)dx ?
5
5
2
0
2
0
 f ( x)dx and  f ( x)dx . What is an expression for  f ( x)dx ?
5
C.
Suppose you know

2
Suppose you know

Suppose you know
Suppose you know
5
f ( x)dx . What is an expression for
 f ( x)dx ?
0
2
5
0
5
2
2
2
 f ( x)dx and  f ( x)dx . What is an expression for  f ( x)dx ?

2
2
5
5
F.

 f ( x)dx ?
0
5
f ( x)dx and
2
E.

f ( x)dx . What is an expression for
2
2
D.
5
2
f ( x)dx and
f ( x)dx and

2
f ( x)dx . What is an expression for
 f ( x)dx ?
0
2
7. The graph of a derivative f (x ) is shown in the figure.
Fill in the table of values for f (x) given that f (0)  2 .
x
f(x)
0
2
1
2
3
4
5
6
*Hint: Don’t forget to ADD the initial condition!
8. Water is leaking out of a tank at a rate of R(t ) gallons/hour, where t is measured in hours.
A. Write a definite integral that expresses the total amount of
water that leaks out in the first two hours.
B. The figure at the right is a graph of R(t ) . Shade the region
whose area represents the total amount of water that leaks out
in the first two hours.
C. If a Right Hand Sum with 8 rectangles, R8 , were used to
estimate the definite integral in Part A, would the result
overestimate or underestimate the exact amount? Explain why.
Use the first part of the FTC to evaluate the definite integrals.
8
9.

1
3
x x  1 dx

10.
1
4
 3 sin(x) dx
11.
0
0
Find the general indefinite integral (don’t forget +C).

1 
dx
12.   5e x  x 
13.    x11 dx
x



x
14.

dx
1
2
dx
1 x2
3