Deck 16: Methods and Applications of Integration

Question 1

Use the formula
+ C to find

Answer 1

( + 1)

Question 2

Use the formula
= -
ln
+ C to find
.

Answer 2

-1 ln + C

Question 3

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Answer 3

Form II, + C

Question 4

Find the present value of a continuous annuity at an annual rate of 6% for two years if the payment at time t is at the annual rate of f(t)= 200 dollars.Express your answer in terms of e.

Question 5

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 6

Use the formula
ln
+ C to find
.

Question 7

Use the table below to determine
Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 8

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 9

Use the formula = -
Ln
+ C to find
)

A) -ln + C
B) -2 ln + C
C) - ln + C
D) -4 ln + C
E) - ln + C

Question 10

Use the formula
= -
ln
+ C to find

Question 11

Use the formula = ln
+ C to find
)

A) ln + C
B) ln + C
C) 3 ln + C
D) ln + C
E) ln + C

Question 12

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 13

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 14

Find the present value of a continuous annuity at an annual rate of 9% for five years if the payment at time t is at the annual rate of f(t)= 3000 dollars.Express your answer in terms of e.

Question 15

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 16

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 17

Use the formula
=
+ C to find
.

Question 18

Use the formula
= ln
+ C to find

Question 19

Use the table below to determine
dx.Indicate the formula number employed.
I.
+ C
II.
+ C
III.

+ C
IV.
+ C

Question 20

Use the tables to find:
dx

Question 21

Use the tables to find:
dx

Question 22

Suppose the points (0,2),(1,3)and (2,0)lie on the graph of the continuous function f.Using the trapezoidal rule and all of these points,an approximation to is

A) 2.
B) 4.
C) 6.
D) 8.
E) 10.

Question 23

Use the trapezoidal rule with n = 3 to estimate the value of

Question 24

Use Simpson's rule with n = 4 to estimate the value of

Question 25

Use Simpson's rule with n = 4 to find an approximate value of

Question 26

Use the tables to find:

Question 27

Use the tables to find:

Question 28

Use Simpson's rule with n = 4 to find an approximate value of

Question 29

Use the tables to find:
dx

Question 30

Use the tables to find:

Question 31

Use the tables to find:

Question 32

Use the trapezoidal rule with n = 4 to estimate the value of

Question 33

Suppose that the points (3,2),(3.25,2.5),(3.5,2.25),(3.75,2)and (4,1.5)lie on the graph of the continuous function f,where f(x)≥ 0.Use Simpson's rule and all of the given points to approximate the area between the graph of f and the x-axis on the interval
.

Question 34

Suppose that the points (-1,2),(-0.5,1),(0,0.5),(0.5,0),and (1,1)lie on the graph of the continuous function f,where f(x)≥ 0.Using Simpson's rule and all of these points,an approximation to the area between the graph of f and the x-axis on the interval is

A) sq units.
B) sq units.
C) sq units.
D) sq units.
E) none of the above

Question 35

Use the tables to find:

Question 36

Suppose that the points (1,1),(2,4),and (3,2)lie on the graph of the continuous function f,where f(x)≥ 0.Use the trapezoidal rule and all of the given points to approximate the area between the graph of f and the x-axis on the interval
.

Question 37

Use the tables to find:

Question 38

Use Simpson's rule with n = 4 to estimate the value of

Question 39

Use the trapezoidal rule with n = 4 to estimate the value of

A)
B) 83
C)
D) 62
E)

Question 40

Use the tables to find:

Question 41

An oil tanker is losing oil at a rate of R'(t)=
where t is the time in minutes and R(t)is the radius of the oil slick in feet.Use n = 6 and Simpson's rule to approximate
the size of the radius after 5 seconds.

Question 42

Find the exact area of the region bounded by y = f(x)=
+ x,x = 0,x = 2,and the x-axis.

Question 43

Use definite integrals to find the area between y = -
,the x-axis,x = -1,and x = 1.

Question 44

Suppose that the accompanying table shows the velocity of a car every second for 8 seconds.Use the Trapezoidal Rule to approximate the distance traveled by the car in the 8 seconds.

A) 145.5 feet
B) 291 feet
C) 162 feet
D) 221.5 feet

Question 45

Find the area of the region bounded by the curve y = f(x)=
- 4x + 3 and the x-axis on the interval
.Use your graphing calculator to see a graph of this area.

Question 46

It is estimated that service industries will grow at the rate of R'(t)= 5
where t is the time in decades and R(t)is the percent of (non-farming)workers who are employed in the service industry.If 30 percent of the (non-farming)workers are now in the service industry,then use the trapezoidal rule and
to approximate 30 +
the percentage of workers who will be working in the service industry in 2 decades.

Question 47

Find the area of the region bounded by the curve y = f(x)=
- x and the x-axis on the interval
.Use your graphing calculator to see a graph of this area.

Question 48

Find the area between the graph of y = f(x)=
,and the x-axis on the interval
.Use your graphing calculator to see a graph of this area.

Question 49

Find the exact area of the region bounded by y =
- 4x and the x-axis.Also sketch the region.

Question 50

Find the area between the graph of y = f(x)=
+
+ 3 and the x-axis on the interval
.Use your graphing calculator to see a graph of this area.

Question 51

The exact area of the region bounded by the graphs of y = + x + 1,x = -2,x = 1,and the x-axis is

A) sq units.
B) sq units.
C) sq units.
D) sq units.
E) sq units.

Question 52

Find the area between the graph of y = f(x)=
,and the x-axis on the interval
.Use your graphing calculator to see a graph of this area.

Question 53

A surveyor measured the length of a piece of land at 100-ft intervals (x),as shown in the table.Use Simpson's Rule to estimate the area of the piece of land in square feet.

A) 24,000
B) 25,000
C) 29,500
D) 24,500

Question 54

Use the trapezoidal rule with n = 3 to find an approximate value of

Question 55

It is estimated that service industries will grow at the rate of R'(t)= 5
where t is the time in decades and R(t)is the percent of (non-farming)workers who are employed in the service industry.If 30 percent of the (non-farming)workers are now in the service industry,then use Simpson's rule and n = 4 to approximate
30 +
the percentage of workers who are working in the service industry in 2 decades.

Question 56

A yeast culture is growing at a rate of A'(t)= 0.3
where t is the time in hours and A(t)is the amount in grams.Use n = 8 and the trapezoidal rule to approximate
the amount the culture grew over the first four hours.

Question 57

The velocity of an object is given by V(t)=
,where t is the time in seconds and V(t)is the speed in feet/sec.Use n = 6 and Simpson's rule to approximate
the distance moved over the interval
.

Question 58

The exact area of the region bounded by the graphs of y = - 4,and the x-axis from x = 0 to x = 4 is

A) sq units.
B) sq units.
C) sq units.
D) 12 sq units.
E) 16 sq units.

Question 59

The velocity of an object is given by V(t)=
,where t is the time in seconds and V(t)is the speed in feet/sec.Use n = 6 and the trapezoidal rule to approximate
the distance moved over the interval
.

Question 60

Find the exact area of the region bounded by y =
+ 1,x = 1,x = 3,and the x-axis.Also sketch the region.

Question 61

Express the area of the shaded region in terms of an integral (or integrals).Do not evaluate the expression:

Question 62

The exact area of the region bounded by the graphs of y = x,y = ,y = 2,and y = 3 is

A) sq units.
B) sq units.
C) sq units.
D) sq units.
E) sq units.

Question 63

Find the area of the region bounded by the given equations:
y =
+ 6x + 10 and y = x + 6

Question 64

Find the area of the region bounded by the curves y =
and y = x + 2 from x = 1 to x = 2.

Question 65

The exact area of the region bounded by the graphs of y = - 5 and y = 2x + 3 is

A) sq units.
B) 36 sq units.
C) 60 sq units.
D) 24 sq units.
E) sq units.

Question 66

The demand equation for a certain product is p = 30 - 0.1q,where p is the price per unit (in dollars)for q units.If its supply equation is p = 0.15q + 5,find the consumers' surplus and the producers' surplus when market equilibrium has been established.

Question 67

The exact area of the region bounded by the graphs of y = x and y = is

A) sq unit.
B) sq unit.
C) sq unit.
D) sq unit.
E) sq unit.

Question 68

The demand equation for a certain product is p = 200 -
,where p is the price per unit (in dollars)for q units.If its supply equation is p = 6q + 160,find the consumers' surplus and the producers' surplus when market equilibrium has been established.

Question 69

Find the area of the region bounded by the curves y =
- 4 and y = 2x +
from x = 1 to x = 2.

Question 70

Find the exact area of the region bounded by the graphs of y = 9 -
and y = 5 - 3x.Also sketch the region.

Question 71

Find the exact area of the region bounded by the graphs of y = -x,y = 2x,y = 1,and y = 2.Also sketch the region.

Question 72

Find the area of the region enclosed by the graphs of y =
-
and y = 2x.

Question 73

The demand equation for a certain product is p = 400 - 2q,where p is the price per unit (in dollars)for q units.If its supply equation is p = q + 100,then the consumers' surplus when market equilibrium is established is

A) $9000.
B) $9500.
C) $10,000.
D) $10,500.
E) $11,000.

Question 74

Find the exact area of the region bounded by the graphs of x =
and x - y - 2 = 0.Also sketch the region.

Question 75

Express the area of the shaded region in terms of an integral (or integrals).Do not evaluate the expression:

Question 76

Find the exact area of the region bounded by y = f(x)= 8 - 2x -
and the x-axis.

Question 77

Find the exact area of the region bounded by the graphs of y = 8 - 2x -
and y = 3x + 2.Also sketch the region.

Question 78

Find the area of the region bounded by the given equations:
y =
- x - 9 and y = -
+ 3x + 7

Question 79

Express the area of the shaded region in terms of an integral (or integrals).Do not evaluate the expression:

Question 80

Find the exact area of the region bounded by y = f(x)=
,x = -2,x = 2,and the x-axis.