AP Calculus
Review Chapter 6
Mathematician:
11/22/13
Chapter 6: The Definite Integral
6.1 Estimating with Finite Sums
____ Use approximation methods LRAM, RRAM, & MRAM to estimate the area under a curve
____ Interpret the applied meaning of the result for area under a curve
6.2 Definite Integrals
____ Write and evaluate a Riemann sum
____ Understand the connection between a Riemann sum and a definite integral
____ Understand the connection of a definite integral to net area
____ Compute the area under a curve using a numerical integration procedure
6.3 Definite Integrals and Antiderivatives
____ Apply properties of definite integrals as areas
____ Apply rules for definite integrals and find the average value of a function
6.4 Fundamental Theorem of Calculus
____ Apply the Fundamental Theorem of Calculus, part 1, to integral functions and graphs
____ Use the Fundamental Theorem of Calculus, part 2, to evaluate definite integrals and
compute enclosed areas.
6.5 Trapezoidal Rule
____ Use the Trapezoidal Rule to estimate the area under a curve
____ Compute error bounds using the formula for Trapezoidal error
Topic: Rules for Integrals
9
9
 f (x )dx
Given:
 f (x )dx
 1
9
 h (x )dx
5
7
1
4
7
1) Evaluate the following integrals…
9
A)
7
 2f (x )  3h (x )dx
B)
7
C)
7
7
 f (x )dx
D)
1
A)
The average value of cos x over the interval
3

B)
1
  f (x )dx
7
Topic: Average Value
2)
 f (x )dx
1
2
C)

3
x
3( 2  3 )


2
is
D)
3
2
E)
2
3
y=f(x)
Given the graph of f ( x )  x  3  2 , find the average
3)
10
9
value of the function on the interval [-4, 6].
8
7
6
5
4
3
2
1
-4 -3 -2 -1
Topic: Using Geometry or the FTC to calculate the exact value of an integral
6

 x  3dx
4)
5)
2

2
 -cos(x )dx
6)
0
8) Use the graph of f(x), which consists of
line segments and semi-circles to answer
the following:
A)
0
 f (x )dx

5
B)
4
f (x )dx

0
C)
6
 f (x )dx
5
 x sin( x )dx


7)
3
 2x  3x dx
1
2
1
2
3
4
5
6
Topic: Net vs. Total Area
3
  4  x dx
9)
2

0
10) Find the total area between the curve y  4  x 2 and the x-axis from 0  x  3 .
6
5
4
3
2
1
-4
-3
-2
-1
-1
1
2
3
4
-2
-3
-4
-5
-6
11)
12)
The graph of the velocity of a particle moving on the x-axis is given. The particle starts at
x = 3 when t = 0.
A)
Find where the particle is at the end
of the trip.
B)
Find the total distance traveled by
the particle.
Given the velocity, v (t )  4t 3  16t 2  12t , where t is measured in seconds.
Find the total distance traveled from 0 to 3 seconds.
Topic: Area Approximations
1
13)
If the trapezoidal rule is used with n = 5, then
dx
1 x
2
is approximately…
0
A)
0.784
14)
15)
B)
C)
D)
E)
1.567
C)
1.959
D)
3.142
E) 7.837
If a Midpoint Riemann sum with four rectangles of equal width is used to approximate the
area enclosed between the x-axis and the graph of y = 4x - x², the approximation is
A) 10
A)
B)
B) 10.5
C) 10.6
D)
10.75
E) 11
A Left Riemann sum with 4 equal subdivisions is used to approximate the area under the sine
curve from x = 0 to x = π. What is the approximation?

  3 
0  

4
4 2 4 

1
3 
 1 
 0  
4
2 2


2
2
1
 0 

4
2
2 

1
2
3

 0  

4
2 2
2 
 1
2
3 

 1 
 
42 2
2

16) GIVEN DATA
Using the following table of data, answer the
following questions.
T
0
1
2
3
4
5
6
(hours)
R(t)
12
28
32
37
42
55
72
17) GIVEN A FUNCTION
Using f (x )  x 2 over the interval [0, 2],
answer the following questions.
(cars
per hr)
A) Use trapezoid rule with 3 subintervals to
approximate the area under the curve from 0
to 6 hours.
A) Use trapezoid rule with 4 subintervals to
approximate the area under the curve from 0
to 2.
B) Use right Riemann sums with 6 subintervals
to approximate the area under the curve from
0 to 6 hours.
B) Use right Riemann sums with 4 subintervals
to approximate the area under the curve from
0 to 2.
C) Use left Riemann sums with 6 subintervals
to approximate the area under the curve from
0 to 6 hours.
C) Use left Riemann sums with 4 subintervals
to approximate the area under the curve from
0 to 2.
D) Use midpoint values with 3 subintervals to
approximate the area under the curve from 0
to 6 hours.
D) Use midpoint values with 2 subintervals to
approximate the area under the curve from 0
to 2.
E) Consider the units. What does the area
under the curve give you?
Topic: Initial Value
18)
A)
A particle moves along a line with acceleration 2 + 6t at time t. When t = 0, its velocity
equals 3 and its position equals 2. Find the position of the particle at t = 1.
2
B)
Topic:
19)
20)
5
C)
6
D)
7
E)
8
FTC Part I
d
dx
2x
 f (t )dt

3
A)
2 f (2x )  f (3) 
B)
f (2x )  f (3)
D)
2f (2x )
E)
2xf (x )
If f (x ) 
3x 2

C)
2xf (2x )
e t  1dt , then f '(1) =
3
A)
21)
28.41
If h (x ) 
B)
x
27.55
C)
18.55
D)
12.41
E)
4.59
1
8
D)
3
5
E)
1
2
 1  2t  dt , then h "(0.5) 
3
0
A)
0
B)
1
12
C)
22)
The graph of a function, f, consists of a semicircle and two line segments as shown below.
graph of f
5
x
Let g ( x) 
 f (t )dt
4
3
1
a)
2
Find g(2)
1
-5 -4 -3 -2 -1
1
2
3
4
5
-1
-2
b)
-3
Find g’(5)
-4
-5
c)
d)
Topic:
23)
Find g’’(3)
Find all values of x on the open interval (-4, 5) at which g has a relative maximum.
Integration Techniques
The graph of f(x) below consists of three equivalent quarter-circles.
Determine whether the following statements are true or false.
A)
B)
C)
2
2
0
0
f (x )dx   f (x )dx
2
4
1
f (x )dx   f (x )dx
2 2
2
4
2
2
0
 f (x )dx  3f (x )dx
24)
26)
4  cos3 x
 cos2 x dx
3
2

2
2
28)
25)
x 4  2x
1 x 2 dx
3
4
5
1  x2
 4x
4
dx
dx
27)

1
29)

dy
y3
x
2

dx
2
x