Deck 6: The Definite Integral

Question 1

Find: dx
Integrate the terms in the order they appear.
Enter your answer as a sum of power functions in standard form with any fractional powers or coefficients reduced of form .

Answer 1

+ + C

Question 2

Find: dx

A) - 4 + C
B) - 4x + C
C) - 4 + C
D) + C
E) none of these

Answer 2

- 4x + C

Question 3

Find: dx

A) - 2 + C
B) - 8 + C
C) - 2 + C
D) - 2 + C

Answer 3

- 2 + C

Question 4

Find all antiderivatives of the function.
f(x) = Enter your answer as a polynomial in x in standard form.

Question 5

Find: Enter terms in the order in which they appear in standard power function forms with any constant at the right end.

Question 6

Find: Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form .

Question 7

Find: dx
Enter terms in the same order in which they appear in the integral.

Question 8

Find: dx
Enter a polynomial in x in standard form with any fractional coefficients or powers in reduced form .

Question 9

Find: Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form .

Question 10

Determine if the function F is the general antiderivative of the function f. F(t) = 5t2 + 3et + C; f(t) = 10t + 3e

Question 11

Find all antiderivatives of the function.
f(y) = Enter your answer as a polynomial in y in standard form.

Question 12

Determine if the function F is the general antiderivative of the function f. F(x) = 5x + C; f(x) = 5

Question 13

Find: dx

A) - + C
B) 6 - ln + C
C) + + C
D) + + C

Question 14

Find: Enter your answer in standard form (use a ).

Question 15

Find: Enter your answer as a power function in x in standard form with any fractional coefficients or powers in reduced form and any constant at the right end.

Question 16

Find: Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form .

Question 17

Find all antiderivatives of the function.
f(x) = Enter your answer with any fractional coefficients and powers in reduced form .

Question 18

Find all antiderivatives of the function.
f(x) = Enter your answer as a polynomial in x in standard form.

Question 19

Find all antiderivatives of the function.
f(x) = Enter your answer as a polynomial in x in standard form.

Question 20

Find: dx
Enter your answer in standard form (no fractions).

Question 21

Find a function f(x) with the following property: f'(x) = $x ^ { 6 }$ + $\frac { x ^ { 4 } } { 3 }$ - $x ^ { 3 }$ + $\frac { 5 } { 3 }$ $x ^ { 2 }$ , f(0) = e.

A) f(x) = 6 $x ^ { 7 }$ + $\frac { 4 } { 3 }$ $x ^ { 5 }$ - $3 x ^ { 4 }$ + $\frac { 10 } { 3 }$ $x ^ { 3 }$
B) f(x) = $\frac { x ^ { 7 } } { 6 }$ + $\frac { x ^ { 5 } } { 12 }$ - $\frac { x ^ { 4 } } { 3 }$ + $\frac { 5 } { 6 }$ $x ^ { 3 }$ + ex
C) f(x) = $\frac { x ^ { 7 } } { 7 }$ + $\frac { x ^ { 5 } } { 15 }$ - $\frac { x ^ { 4 } } { 4 }$ + $\frac { 5 } { 9 }$ $x ^ { 3 }$ + e
D) $f(x)=7^{e^{7}}+\frac{5}{3 \mathrm{e}^{5}}-4^{e^{4}}+5^{e^{3}}$

Question 22

A ball is thrown upward with initial velocity of 144 feet per second. How high will the ball go? (Recall that from physics, it is known that the velocity at time t is feet per second.)Enter just an integer (no units).

Question 23

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6

A) = 15.75
B) = 21.75
C) = 12.75
D) = 8.25

Question 24

Find: Enter your answer using standard power function form (a ), leaving the terms in the order in which they appear in the integral.

Question 25

Suppose f(x) is an antiderivative of $\frac { 2 } { \sqrt { x } }$ and f(0) = 1. What is f(9)?

A) 13
B) - $\frac { 4 } { 3 }$
C) 1
D) -4
E) none of these

Question 26

Find all functions f(x) with the following property: f'(x) = 3 + 2x + 1.
Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form .

Question 27

Find the value of k that makes the antidifferentiation formula true.
= k + C

A) 6
B) -6
C)
D) -

Question 28

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints.
Enter just an integer.
f(x) = 2x + 1; 1 ≤ x ≤ 5, n = 4

Question 29

A company estimates that its marginal profit for producing a product is given by where is in dollars per unit. Given that when find the total profit realized from selling 300 units of product.

A) $12,100
B) $7700
C) $6700
D) $4100

Question 30

Find the value of k that makes the antidifferentiation formula true.
= k ln|7 - x| + C

A) -1
B)
C) -
D) 1

Question 31

Find: Enter your terms in the same order in which they appear in the integral with each term in standard power function form; any fractions reduced form .

Question 32

A newspaper is launching a new advertising campaign in order to increase the number of daily subscribers. The newspaper currently (t = 0) has 26,000 daily subscribers and management expects that number, S(t), to grow at the rate of subscribers per day, where t is the number of days since the campaign began. How long (to the nearest day) should the campaign last if the newspaper wants the number of daily subscribers to grow to 49,000?

A) 69 days
B) 57 days
C) 33 days
D) 44 days

Question 33

Find a function f(x) with the following property: f'(x) = 2 - + , f(1) = -2.

A) f(x) = - 4 ln|x| - - ( - 1)
B) f(x) = - 4 ln|x| - + 3 + 4e
C) f(x) = - 4x - +
D) f(x) = - 4 ln|x| - -

Question 34

Determine Δx formed by partitioning the given interval into n subintervals.
-8 ≤ x ≤ -4; n = 6

A) Δx =
B) Δx =
C) Δx = -
D) Δx = 4

Question 35

A rock is dropped from a balloon hovering at 4800 ft above the ground. Its velocity at time t seconds is feet per second. Find how long it takes for the rock to reach the ground.
Enter just a real number to one decimal place (no units).

Question 36

Find a function f(x) with the following property: f'(x) = - , f(2) = 0.

A) f(x) = +
B) f(x) = ln +
C) f(x) = ln - 2
D) f(x) = -ln + 2
E) none of these

Question 37

Find all functions f(x) with the following property: f'(x) = + 2 - + 6.
Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form .

Question 38

Find: Enter your terms in standard forms in the order in which they appear in the integral.

Question 39

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6

A) = 12.75
B) = 15.75
C) = 8.25
D) = 21.75

Question 40

Find: .
Enter using standard power function form a , with any fractions reduced of form .

Question 41

Calculate.

A) ( - )
B) - + C
C) -
D) ( - )
E) none of these

Question 42

Given f(x) = lnx on the interval 1 ≤ x ≤ 5 and with n = 2, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all real numbers rounded to two decimal places and separated by commas.
Enter the numbers in the order that answers (a), (b), (c) but do not label.

Question 43

Enter just a reduced fraction of form .

Question 44

Draw the region whose area is given by the definite integral. dx

A)
B)
C)
D)
Calculate.

Question 45

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints.
Enter just a real number to two decimal places.
f(x) = ; 0 ≤ x ≤ 2 , n = 4

Question 46

Draw the region whose area is given by the definite integral. dx

A)
B)
C)
D)

Question 47

Given the graph of the function y = , set up the definite integral that gives the area of the shaded region.

A) dx
B) dx
C) dx
D) dx

Question 48

Calculate.

A) -e + ln 3
B) ln 3
C) -ln 3
D)
E) none of these

Question 49

Given f(x) = + x +1 on the interval 0 ≤ x ≤ 4 and with n = 5, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all integers separated by commas.
Enter the numbers in the order that answers (a), (b), (c) but do not label. Round to the nearest whole number.

Question 50

Given the graph of the function y = + 8, set up the definite integral that gives the area of the shaded region.

A) dx
B) dx
C) dx
D) dx

Question 51

Given f(x) = - 1 on the interval 1 ≤ x ≤ 5 and with n = 4, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all integers separated by commas.
Enter the numbers in the order that answers (a), (b), (c) but do not label.

Question 52

Given set up a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. Is the following the correct sum?

Question 53

Draw the region whose area is given by the definite integral. dx

A)
B)
C)
D)

Question 54

Given set up a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the midpoints. Is the following the correct answer?

Question 55

Use a Riemann sum to approximate the area under the graph of Use the midpoints of the intervals.
Enter your answer as just a reduced fraction of form .

Question 56

Use a Riemann sum to approximate the area under the graph of Use the left endpoints of the interval.
Enter an unlabeled answer.

Question 57

Use a Riemann sum to approximate the area under the graph of Use the right endpoints.
Enter just a real number to two decimal places.

Question 58

Draw the region whose area is given by the definite integral. dx

A)
B)
C)
D)

Question 59

Calculate.

A) +
B) e +
C) - e -
D)
E) none of these

Question 60

Calculate.

A)
B) -
C) -
D) -
E) none of these

Question 61

What is the area under the curve y = $\frac { 2 } { x }$ between x = 1 and x = 3?

A) ln 9
B) $\frac { 1 } { 2 }$ ln 3
C) 2 ln 3 - e
D) 2 ln 3 - 2
E) none of these

Question 62

Enter your answer as a(b + ) with any fractions in reduced form .

Question 63

Enter a reduced fraction of form .

Question 64

Determine the area under the curve y = from x = 1 to x = e.
Enter just an integer.

Question 65

Enter just an integer.

Question 66

Enter your answer as a( - ) + d.

Question 67

What is the area under the curve y = $x ^ { 3 }$ + x from x = 1 to x = 2?

A) $\frac { 21 } { 2 }$
B) $\frac { 6 } { 4 }$
C) $6 \frac { 1 } { 4 }$
D) $\frac { 21 } { 4 }$
E) none of these

Question 68

Enter just an integer.

Question 69

Enter just a reduced fraction of form .

Question 70

Enter a reduced fraction of form .

Question 71

dx
Enter just a real number (no approximations).

Question 72

Determine the area under the curve y = 4x + 4 from x = 2 to x = 3.
Enter an integer.

Question 73

What is the area under the curve f(x) = 3 $x ^ { 2 }$ + 2x + 1 from x = - $\frac { 3 } { 2 }$ to x = - $\frac { 1 } { 2 }$ ?

A) $\frac { 9 } { 4 }$
B) $\frac { 51 } { 4 }$
C) $\frac { 15 } { 4 }$
D) - $\frac { 15 } { 4 }$
E) none of these

Question 74

Enter your answer as a( ± ). Any fractions in reduced form .

Question 75

Enter your answer as a reduced fraction of form .

Question 76

What is the area under the curve y = + 3 between x = - and x = ?

A) - e
B) e +
C)
D)
E) none of these

Question 77

What is the area under the curve y = between x = 1 and x = 2?

A) 2 - 2
B)
C) ln
D)
E) none of these

Question 78

Enter just an integer.

Question 79

Find the area under the curve y = - 2x from x = -3 to x = -2.
Enter a ± ln b using reduced fractions of form and integers.

Question 80

Enter your answer as a( - ).