Deck 4: Definite Integrals

Question 1

Write the following sum in sigma notation.2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

A) $\sum _ { k = 1 } ^ { 6 } 2$
B) $\sum _ { k = 1 } ^ { 8 } 2$
C) $\sum _ { k = 1 } ^ { 9 } 2$
D) $\sum _ { k = 1 } ^ { 7 } 2$

Answer 1

$\sum _ { k = 1 } ^ { 9 } 2$

Question 2

Write the following sum in sigma notation. $$\frac { 1 } { 8 } + \frac { 1 } { 27 } + \frac { 1 } { 64 } + \frac { 1 } { 125 } + \frac { 1 } { 216 }$$

A) $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 2 } }$
B) $\sum _ { k = 2 } ^ { 7 } \frac { 1 } { k ^ { 3 } }$
C) $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { 2 k ^ { 2 } }$
D) $\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 3 } }$

Answer 2

$\sum _ { k = 2 } ^ { 6 } \frac { 1 } { k ^ { 3 } }$

Question 3

Write the following sum in sigma notation.11 + 18 + 27 + 38 + 51 + 66

A) $\sum _ { k = 2 } ^ { 7 } k ^ { 2 } + 2$
B) $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 2$
C) $\sum _ { k = 2 } ^ { 8 } k ^ { 2 } + 2$
D) $\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 3$

Answer 3

$\sum _ { k = 3 } ^ { 8 } k ^ { 2 } + 2$

Question 4

Write the following sum in sigma notation. $$2 + \frac { 3 } { 4 } + \frac { 4 } { 9 } + \frac { 5 } { 16 } + \frac { 6 } { 25 } + \frac { 7 } { 36 } + \frac { 8 } { 49 }$$

A) $\sum _ { k = 3 } ^ { 9 } \frac { k + 1 } { k }$
B) $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k }$
C) $\sum _ { k = 1 } ^ { 7 } \frac { k + 1 } { k ^ { 2 } }$
D) $\sum _ { k = 3 } ^ { 10 } \frac { k + 1 } { k ^ { 2 } }$

Question 5

Write the following sum in sigma notation. $$2 + \frac { 5 } { 3 } + \frac { 6 } { 4 } + \frac { 7 } { 5 } + \frac { 8 } { 6 } + \frac { 9 } { 7 } + \frac { 10 } { 8 } + \frac { 11 } { 9 }$$

A) $\sum _ { k = 2 } ^ { 9 } \frac { k + 1 } { k }$
B) $\sum _ { k = 2 } ^ { 8 } \frac { k + 2 } { k }$
C) $\sum _ { k = 2 } ^ { 9 } \frac { k + 3 } { k }$
D) $\sum _ { k = 2 } ^ { 9 } \frac { k + 2 } { k }$

Question 6

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 2 } ^ { 7 } k ^ { 2 }$$

Question 7

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 1 } ^ { 5 } ( 3 + k ) ^ { 2 } + 1$$

Question 8

Write out each sum in expanded form, and then calculate the value of the sum. $$\sum _ { k = 1 } ^ { 6 } \ln k$$

Question 9

Find a formula for the sum and then use it to calculate the sum for n = 100, n = 500, and n = 1000. $$\sum _ { k = 1 } ^ { n } ( 2 - k )$$

A) $\frac { 3 n + n ^ { 2 } } { 2 }$
B) $\frac { 4 n - n ^ { 2 } } { 2 }$
C) $\frac { 3 n - n ^ { 2 } } { 2 }$
D) $\frac { 2 n - n ^ { 2 } } { 2 }$

Question 10

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } - 2 k + 3 \right)$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Question 11

Find a formula for the sum $\sum _ { k = 2 } ^ { n } ( 2 + k ) ^ { 2 }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Question 12

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \frac { \left( k ^ { 2 } + 2 k + 2 \right) } { n ^ { 3 } }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Question 13

Find a formula for the sum $\sum _ { k = 1 } ^ { n } \frac { k ^ { 3 } + 1 } { 2 }$ and then use it to calculate the sum for n = 100, n = 500, and n = 1000.

Question 14

Write the given expression in one sigma notation (with some extra terms added or subtracted from the sum, as necessary). $2 \sum _ { k = 3 } ^ { 20 } k ^ { 2 } + 3 \sum _ { k = 2 } ^ { 21 } k + \sum _ { k = 1 } ^ { 24 } 1$

Question 15

Find $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } - 2 k + 3 } { n ^ { 3 } }$

A) $\frac { 1 } { 5 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 2 } { 5 }$
D) $\frac { 1 } { 3 }$

Question 16

Find $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } + 2 k + 3 } { 2 n ^ { 4 } }$

A) $\frac { 1 } { 3 }$

B) $\frac { 1 } { 2 }$

C) $\frac { 1 } { 6 }$

D) $\frac { 1 } { 4 }$

Question 17

Approximate the area between the graph of $f ( x ) = x ^ { 2 } + 4 x$ and the x-axis on the interval [2, 4], using a right sum with (a) n = 4, (b) n = 6.

Question 18

Approximate the area between the graph of $f ( x ) = x ^ { 2 } + 4 x$ and the x-axis on the interval [2, 4], using a left sum with (a) n = 4, (b) n = 6.

Question 19

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a right sum with (a) n = 2, (b) n = 4.

Question 20

Approximate the area between the graph of $f ( x ) = ( x - 2 ) ^ { 2 } + 1$ and the x-axis on the interval [1, 5], using a left sum with (a) n = 2, (b) n = 4.

Question 21

Approximate the area between the graph of $f ( x ) = 9 - x ^ { 2 }$ and the x-axis on the interval [0, 3], for n = 6, using (a) left sums (b) right sums.

Question 22

Approximate the area between the graph of $f ( x ) = \sqrt { x + 1 }$ and the x-axis on the interval [1, 3], for n = 4, using (a) left sums (b) right sums.

Question 23

Use geometry to find the exact value of $\int _ { 0 } ( 3 - x ) d x$

A) 5
B) 3
C) 6
D) 4

Question 24

Use geometry to find the exact value of $\int _ { - 1 } | 2 x + 1 | d x$

A) 20.25
B) 20
C) 20.5
D) 19.75

Question 25

Use geometry to find the exact value of $\int _ { - 2 } ^ { 6 } 10 d x$

A) 60
B) 80
C) 100
D) 70

Question 26

Use geometry to find the exact value of $\int _ { - 2 } ^ { 2 } \sqrt { 4 - x ^ { 2 } } d x$

A) $\pi$
B) $\pi$ /2
C) 4 $\pi$
D) 2 $\pi$

Question 27

Use geometry to find the exact value of $\int _ { - 1 } ^ { 4 } ( 2 - | x - 2 | ) d x$

A) 9/2
B) 5/2
C) 7/2
D) 11/2

Question 28

Use geometry to find the exact value of $\int _ { 0 } 2 + \sqrt { 9 - x ^ { 2 } } d x$

A) $\frac { 6 + 81 \pi } { 2 }$
B) $\frac { 6 + 9 \pi } { 4 }$
C) $\frac { 6 + 81 \pi } { 4 }$
D) $\frac { 24 + 9 \pi } { 4 }$

Question 29

If $\int _ { - 1 } ^ { 4 } f ( x ) d x = 3$ , $\int _ { - 3 } ^ { 4 } f ( x ) d x = 6$ , $\int _ { - 1 }^4 g ( x ) d x = - 2$ and $\int _ { 4 } ^ { 6 } g ( x ) d x = 5$ , find the following, if possible.
(a) $\int _ { - 3 } ^ { - 1 } f ( x ) d x$
(b) $\int _ { - 1 }^6 ( g ( x ) + 3 ) d x$
(c) $\int _ { - 1 } ^ { 4 } ( 2 f ( x ) + x ) d x$

Question 30

If $\int _ { - 1 } ^ { 4 } f ( x ) d x = 3$ , $\int _ { - 3 } ^ { 4 } f ( x ) d x = 6$ , $\int _ { - 1 }^4 g ( x ) d x = - 2$ and $\int _ { 4 } ^ { 6 } g ( x ) d x = 5$ , find the following, if possible.
(a) $\int _ { 4 } ^ { - 1 } 3 f ( x ) d x$ (b) $\int _ { - 3 } ^ { 4 } ( f ( x ) ) ^ { 2 } d x$ (c) $\int _ { 4 } ^ { 6 } ( 3 g ( x ) - x ) d x$

Question 31

If $\int _ { - 1 } ^ { 4 } f ( x ) d x = 3$ , $\int _ { - 3 } ^ { 4 } f ( x ) d x = 6$ , $\int _ { - 1 } g ( x ) d x = - 2$ and $\int _ { 4 } ^ { 6 } g ( x ) d x = 5$ , find the following, if possible.
(a) $\int _ { - 1 } ^6g ( x ) d x$ (b) $\int _ { 4 } ^ { - 3 } - 2 f ( x ) d x$ (c) $\int _ { - 1 } ^ { 4 } 2 f ( x ) g ( x ) d x$

Question 32

Find $\int _ { 1 }^4 ( 4 - x ) d x$ using Riemann sums (a) using the right sum, (b) using the left sum.

Question 33

Find $\int _ { 0 }^2 x ^ { 2 } d x$ using Riemann sums (a) using the right sum, (b) using the left sum.

Question 34

Find $\int _ { - 1 } ^1( 2 x + 3 ) d x$ using Riemann sums (a) using the right sum, (b) using the left sum.

Question 35

Find $\int \frac { 2 x + 1 } { \sqrt { x } } d x$

A) $\frac { 4 } { 3 } x ^ { 2 / 3 } + 2 x ^ { 1 / 2 } + C$
B) $\frac { 4 } { 3 } x ^ { 2 / 3 } + 2 x + C$
C) $\frac { 2 } { 3 } x ^ { 3 / 2 } + 2 x ^ { 1 / 2 } + C$
D) $\frac { 4 } { 3 } x ^ { 3 / 2 } + 2 x ^ { 1 / 2 } + C$

Question 36

Find $\int \left( 3 - \frac { 2 } { x ^ { 2 } } \right) d x$

A) $3 x + \frac { 2 } { x ^ { 3 } } + C$
B) $3 x - \frac { 2 } { x ^ { 3 } } + C$
C) $3 x + \frac { 2 } { x } + C$
D) $3 x - \frac { 2 } { x } + C$

Question 37

Find $\int 2 \tan ^ { 2 } x d x$

A) $2 \sec x - 2 x + C$
B) $\frac { 2 } { 3 } \tan ^ { 3 } x + C$
C) $2 \tan x + 2 x + C$
D) $2 \tan x - 2 x + C$

Question 38

Find $\int \left( x ^ { 4 } + 2 \right) ^ { 2 } d x$

A) $\frac { 1 } { 9 } x ^ { 9 } + \frac { 2 } { 5 } x ^ { 5 } + 4 x + C$
B) $\frac { 1 } { 9 } x ^ { 9 } + \frac { 4 } { 5 } x ^ { 5 } + 4 x + C$
C) $\frac { 1 } { 6 } x ^ { 6 } + \frac { 2 } { 5 } x ^ { 5 } + 4 x + C$
D) $\frac { 1 } { 9 } x ^ { 9 } + \frac { 4 } { 5 } x ^ { 5 } + 2 x + C$

Question 39

Find $\int \left( 6 x ^ { 2 } + 5 - 2 e ^ { x } \right) d x$

A) $2 x ^ { 3 } + 5 - 2 e ^ { x } + C$
B) $x ^ { 3 } + 5 x - 2 e ^ { x } + C$
C) $2 x ^ { 3 } + 5 x - 2 e ^ { x } + C$
D) $2 x ^ { 3 } + 5 x + 2 e ^ { x } + C$

Question 40

Find $\int \frac { \left( x ^ { 3 } + 2 x \right) ^ { 2 } } { x ^ { 3 } } d x$

A) $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 x + C$

B) $\frac { 1 } { 4 } x ^ { 4 } + x ^ { 2 } + 4 \ln | x | + C$

C) $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + \frac { 4 } { x ^ { 2 } } + C$

D) $\frac { 1 } { 4 } x ^ { 4 } + 2 x ^ { 2 } + 4 \ln | x | + C$

Question 41

Find $\int \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 5 } } d x$

A) $\ln | x | - \frac { 2 } { x } - \frac { 1 } { x ^ { 4 } } + C$
B) $- \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } - \frac { 20 } { x ^ { 6 } } + C$
C) $\ln | x | - \frac { 2 } { x ^ { 2 } } - \frac { 1 } { x ^ { 4 } } + C$
D) $\ln | x | + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x ^ { 4 } } + C$

Question 42

Evaluate $\int _ { - 2 } \left( 4 x ^ { 3 } + 2 x + 1 \right) d x$

A) 15
B) -12
C) -15
D) 12

Question 43

Evaluate $\int _ { 1 } 5 \left( 3 ^ { x } \right) d x$

A) $\frac { 100 } { \ln 3 }$
B) $120 \ln 3$
C) $100 \ln 3$
D) $\frac { 120 } { \ln 3 }$

Question 44

Evaluate $\int _ { 0 } ^ { 2 } \left( x ^ { 2 } - 2 \right) ^ { 2 } d x$

A) 56/13
B) 53/15
C) 56/15
D) 55/16

Question 45

Evaluate $\int _ { 0 } ^ { 1 / 2 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$

A) $\pi$ /3
B) $\pi$ /4
C) $\pi$
D) $\pi$ /6

Question 46

Evaluate $\int _ { 0 } ^ { \pi / 2 } ( 2 \sin x - 3 \cos x ) d x$

A) 1
B) 0
C) -1
D) $1 / 2$

Question 47

Evaluate $\int _ { 0 } ^ { 1 } \frac { e ^ { 3 x } - x e ^ { 2 x } } { e ^ { 2 x } } d x$

A) $e - \frac { 1 } { 2 }$
B) $e - \frac { 3 } { 2 }$
C) $e + \frac { 1 } { 2 }$
D) $e + \frac { 3 } { 2 }$

Question 48

Evaluate $\int _ { 1 } ^ { 3 } \frac { 2 } { x } d x$

A) ln6
B) ln3
C) ln9
D) 1

Question 49

Evaluate $\int _ { 1 } 2 e ^ { x + 3 } d x$

A) $2 e ^ { 4 } \left( e ^ { 2 } + 1 \right)$
B) $2 e ^ { 3 } \left( e ^ { 2 } + 1 \right)$
C) $2 e ^ { 3 } \left( e ^ { 2 } - 1 \right)$
D) $2 e ^ { 4 } \left( e ^ { 2 } - 1 \right)$

Question 50

Evaluate $\int _ { 1 } ^ { 4 } \frac { 3 \sqrt { x } - 2 } { \sqrt { x } } d x$

A) 4
B) 5
C) 3
D) -3

Question 51

Evaluate $\int _ { 1 } ^ { 2 } \frac { \left( x ^ { 2 } + 2 \right) ^ { 2 } } { x ^ { 3 } } d x$

Question 52

Evaluate $\int _ { - 1 } ^ { 4 } | x - 3 | d x$

A) 15/2
B) 9/2
C) 17/2
D) 19/2

Question 53

Evaluate $\int _ { - 1 } ^ { 4 } \left| 9 - x ^ { 2 } \right| d x$

A) 20
B) 15
C) 12
D) 30

Question 54

Evaluate $\int _ { \pi / 3 } ^ { 3 \pi / 2 } | \sin x | d x$

A) 2
B) 5/2
C) 3/2
D) 1

Question 55

Evaluate $\int _ { - 1 } \left| e ^ { 3 x } - 1 \right| d x$

Question 56

Given $f ( x ) = 3 x ^ { 2 }$ over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Question 57

Given $f ( x ) = 4 - x$ over the interval [1, 6], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 1 to x = 6.

Question 58

Given $f ( x ) = 4 - x ^ { 2 }$ over the interval [0, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 4.

Question 59

Given $f ( x ) = x ^ { 2 } + x - 6$ over the interval [-3, 4], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -3 to x = 4.

Question 60

Given $f ( x ) = \sin x$ over the interval [0, 2 $\pi$ ], find:
(a) the signed area,
(b) the absolute area, between the graph of f and the x-axis from x = 0 to x = 2 $\pi$

Question 61

Given $f ( x ) = 1 - e ^ { x }$ over the interval [-1, 2], find: (a) the signed area, (b) the absolute area, between the graph of f and the x-axis from x = -1 to x = 2.

Question 62

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = x + 6$ from x = -3 to x = 0.

A) 10
B) 62/6
C) 10.5
D) 61/6

Question 63

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x ^ { 2 }$ and $g ( x ) = x + 6$

A) 21
B) 123/6
C) 125/6
D) 124/6

Question 64

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x - 2$ and $g ( x ) = x ^ { 2 } - 3 x - 2$ from x = -1 to x = 4.

A) 9
B) 13
C) 10
D) 11

Question 65

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x + 2$ , $g ( x ) = 5 - x$ , and the x-axis from x = 0 to x = 6.

A) 45/4
B) 11
C) 43/4
D) 45/2

Question 66

Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = \sin x$ and $g ( x ) = \cos x$ from x = 0 to x = $\pi$ /2.

A) $\sqrt { 2 } - 2$
B) $2 \sqrt { 2 } - 2$
C) $\sqrt { 2 } + 2$
D) $2 \sqrt { 2 } + 2$

Question 67

Find the exact average value of $f ( x ) = x - 2$ from x = -1 to x = 3.

A) -4
B) -2
C) -1
D) 2

Question 68

Find the exact average value of $f ( x ) = 9 - x ^ { 2 }$ from x = -3 to x = 3.

A) 0
B) 36
C) 18
D) 6

Question 69

Find the exact average value of $f ( x ) = \sin x + \cos x$ from x = - $\pi$ to x = $\pi$

A) $1 / 2$
B) 0
C) 3/2
D) 1

Question 70

Find the exact average value of $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ from x = -3 to x = 0.

A) 3
B) 0
C) 1
D) 2

Question 71

Find the exact average value of $f ( x ) = 3 + \sqrt { x }$ from x = 1 to x = 4.

Question 72

Find the exact average value of $f ( x ) = 2 e ^ { x }$ from x = -1 to x = 3.

Question 73

Given $f ( x ) = 2 x + 1$ , find a number c on the interval (1, 3) such that f(c) is the average value of f on [1, 3].

Question 74

Given $f ( x ) = 16 - x ^ { 2 }$ , find a number c on the interval (0, 4) such that f(c) is the average value of f on [0, 4].

Question 75

Given $f ( x ) = ( x + 1 ) ^ { 2 } - 2$ , find a number c on the interval (-3, 0) such that f(c) is the average value of f on [-3, 0].

Question 76

Given $f ( x ) = 3 + \sqrt { x }$ , find a number c on the interval (1, 4) such that f(c) is the average value of f on [1, 4].

Question 77

Find $\frac { d } { d x } \int _ { 1 } ^ { x } \frac { \cos ^ { 2 } t } { t } d t$

Question 78

Find $\frac { d } { d x } \int _ { x } ^ { 3 } 2 e ^ { - t ^ { 2 } + 3 } d t$

Question 79

Find $\frac { d } { d x } \int _ { 1 } ^ { x ^ { 2 } } \ln t d t$

Question 80

Find $\frac { d } { d x } \int _ { 2 } ^ { \tan x } 3 t ^ { 3 } d t$