Deck 6: The Integral

Question 1

Let A denote the area enclosed by the graph $f ( x ) = 10 - x,$ the x-axis, and the lines $x = 3$ and $x = 5$ . Graphing the region and using plane geometry, we can find that A is

A)8
B)10
C)11
D)12
E)14

Answer 1

12

Question 2

Let A denote the area enclosed by the graph $f ( x ) = | x |,$ the x-axis, and the lines $x = - 2$ and $x = 0$ . Graphing the region and using plane geometry, we can find that A is

A)1
B)1.5
C)2
D)2.5
E)3

Answer 2

2

Question 3

Let A denote the area enclosed by the graph $f ( x ) = | x | \text {, }$ the x-axis, and the lines $x = - 1$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

A)0.5
B)1
C)1.5
D)2
E)2.5

Answer 3

1

Question 4

Let A denote the area enclosed by the graph $f ( x ) = \sqrt { 1 - x ^ { 2 } },$ the x-axis, and the lines $x = 0$ and $x = 1$ . Graphing the region and using plane geometry, we can find that A is

A) $\frac { \pi } { 4 }$
B) $\frac { \pi } { 3 }$
C) $\frac { \pi } { 2 }$
D) $\pi$
E) $2 \pi$

Question 5

Let A denote the area enclosed by the graph $f ( x ) = 3 + \sqrt { 4 - x ^ { 2 } },$ the lines y = 3, x = 0, and x = 2. Graphing the region and using plane geometry, we can find that A is

A) $2 \pi - 3$
B) $2 \pi + 3$
C) $\pi - 3$
D) $\pi + 3$
E) $\pi$

Question 6

Let A denote the area enclosed by the graph $f ( x ) = 4 - | x |,$ the x-axis, and the lines x = -2 and x = 0. Graphing the region and using plane geometry, we can find that A is

A)2
B)3
C)4
D)5
E)6

Question 7

Let A denote the area enclosed by the graph $f ( x ) = ( x - 1 ) ^ { 2 },$ the x-axis, and the lines x = 2 and x = 9. Graphing the region and using plane geometry, we can find that A is

A)
$\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } - 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$
B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 1 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$
C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 7 k } { n } + 2 \right) ^ { 2 } \left( \frac { 7 } { n } \right)$
D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 2 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$
E) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 9 k } { n } + 1 \right) ^ { 2 } \left( \frac { 9 } { n } \right)$

Question 8

Let A denote the area enclosed by the graph $( x ) = \frac { 1 } { x + 1 }$ , the x-axis, and the lines x = 1 and x = 5. Graphing the region and using plane geometry, we can find that A is

A) $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$
B) $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 4 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$
C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 5 } { n } \right)$
D) $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { 4 k } + 2 \right) \left( \frac { 4 } { n } \right)$
E) $\lim _ { n \rightarrow \infty } \sum _ { k = 0 } ^ { n - 1 } \left( \frac { 1 } { \frac { 5 k } { n } + 1 } \right) \left( \frac { 4 } { n } \right)$

Question 9

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$ the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k - 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 2 } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$
D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
E) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$

Question 10

Let A denote the area enclosed by the graph $f ( x ) = \frac { \sqrt { x } } { 3 },$ the x-axis, and the lines x = 0 and x = 9. Graphing the region and using plane geometry, we can find that A is

A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 3 } { n } \sqrt { \frac { k - 1 } { n } }$
B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 27 } { n } \sqrt { \frac { k } { n } }$
C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 3 } { n } \sqrt { \frac { k } { n } }$
D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 9 } { n } \sqrt { \frac { k - 1 } { n } }$
E) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 27 } { n } \sqrt { \frac { k - 1 } { n } }$

Question 11

Suppose S₄ is the lower sum of the area enclosed by the graph $f ( x ) = 2 x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S₄ is

A)26
B)28
C)32
D)48
E)60

Question 12

Suppose S₄ is the upper sum of the area enclosed by the graph $f ( x ) = 2 x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 4 by partitioning [0, 4] into four subintervals [0, 1], [1, 2], [2, 3], and [3, 4]. Then S₄ is

A)26
B)28
C)32
D)48
E)60

Question 13

Suppose S₄ is the upper sum of the area enclosed by the graph $( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S₄ is

A)6
B) $\frac { 9 } { 8 }$
C) $\frac { 11 } { 6 }$
D)2
E) $\frac { 13 } { 6 }$

Question 14

Suppose S₃ is the lower sum of the area enclosed by the graph $( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = 4 by partitioning [1, 4] into three subintervals [1, 2], [2, 3], and [3, 4]. Then S₃ is

A) $\frac { 13 } { 12 }$
B) $\frac { 9 } { 8 }$
C) $\frac { 11 } { 6 }$
D)2
E) $\frac { 13 } { 6 }$

Question 15

Suppose S₄ is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + 2,$ the x-axis, and the lines x = 0 and x = 2 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S₄ is

A)8.5
B)8.25
C)8.125
D)7.825
E)7.75

Question 16

Suppose S₄ is the lower sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 2] into four subintervals [0, 1/2], [1/2, 1], [1, 3/2], and [3/2, 2]. Then S₄ is

A)5.35
B)5.45
C)5.55
D)5.65
E)5.75

Question 17

Suppose S₅ is the upper sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is

A)260
B)340
C)420
D)480
E)500

Question 18

Suppose S₅ is the lower sum of the area enclosed by the graph $f ( x ) = x ^ { 2 } + x,$ the x-axis, and the lines x = 0 and x = 10 by partitioning [0, 10] into five subintervals [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. Then S₅ is

A)240
B)260
C)280
D)300
E)320

Question 19

Suppose S₆ is the upper sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S₆ is

A)23.125
B)23.225
C)23.235
D)23.245
E)23.255

Question 20

Suppose S₆ is the lower sum of the area enclosed by the graph $f ( x ) = 10 - x ^ { 2 },$ the x-axis, and the lines x = 0 and x = 3 by partitioning [0, 3] into six subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3]. Then S₆ is

A)18.625
B)18.635
C)18.645
D)18.655
E)18.665

Question 21

The value of the definite integral $\int _ { - 1 } ^ { 2 } \pi ^ { 2 } d x$ is

A) $3 \pi ^ { 2 }$
B) $5 \pi ^ { 2 }$
C) $7 \pi ^ { 2 }$
D) $9 \pi ^ { 2 }$
E) $11 \pi ^ { 2 }$

Question 22

The value of the definite integral $\int _ { - 4 } ^ { 6 } \frac { 1 } { e } d x$ is

A) $\frac { 2 } { e }$
B) $\frac { 4 } { e }$
C) $\frac { 6 } { e }$
D) $\frac { 8 } { e }$
E) $\frac { 10 } { e }$

Question 23

The value of the definite integral $\int _ { 0 } ^ { 2 } \log _ { 2 } 32 d x$ is

A)8
B)10
C)16
D)32
E)64

Question 24

The value of the definite integral $\int _ { - \pi } ^ { \pi } \frac { e ^ { 3 } - e ^ { - 3 } } { 2 } d x$ is

A)
$\left( \frac { e ^ { 4 } } { 8 } - \frac { e ^ { - 2 } } { 4 } \right) \pi$
B) $\left( \frac { e ^ { 4 } } { 4 } - \frac { e ^ { - 2 } } { 2 } \right) \pi$
C) $\left. e ^ { 3 } - e ^ { - 3 } \right) \pi$
D) $\left( \frac { e ^ { 3 } } { 2 } - \frac { e ^ { - 3 } } { 2 } \right) \pi$
E) $2 \pi \left( e ^ { 3 } - e ^ { - 3 } \right)$

Question 25

The value of the definite integral $\int _ { e ^ { - 1 } } ^ { e } e ^ { \pi } d x$ is

A) $e ^ { \pi } - e ^ { - \pi }$
B) $e ^ { \pi e } - e ^ { - \pi \theta ^ { - 1 } }$
C) $e ^ { 1 + \pi } - e ^ { 1 - \pi }$
D) $e ^ { \pi + 1 } - e ^ { \pi - 1 }$
E) $\frac { \mathrm { e } ^ { \pi + 1 } } { \pi + 1 } \left( e - e ^ { - 1 } \right)$

Question 26

The value of the definite integral $\int _ { - 2 } ^ { 2 } \ln e d x$ is

A)2
B) $\frac { 4 } { e }$
C)ln 4
D) $\ln 2 - \ln \frac { 1 } { 2 }$
E)4

Question 27

If $\int _ { - 2 } ^ { 3 } f ( x ) d x = \frac { 3 } { 2 },$ then $\int _ { - 3 } ^ { 2 } 8 f ( x ) d x$ is

A)-12
B)12
C) $16 / 3$
D)-16/3
E)Impossible to determine

Question 28

If $\int _ { - 1 } ^ { 1 } f ( x ) d x = 6,$ then $\int _ { 0 } ^ { 1 } 2 f ( x ) d x$ is

A)-12
B)-6
C)6
D)12
E)Impossible to determine

Question 29

If $\int _ { 2 } ^ { 1 } 5 f ( x ) d x = 5,$ then $\int _ { 1 } ^ { 2 } 2 f ( x ) d x$ is

A)-5
B)-2
C)2
D)5
E)Impossible to determine

Question 30

If $\int _ { 0 } ^ { 1 } f ( x ) d x = \pi,$ then $\frac { \pi ^ { 2 } } { \int _ { 1 } ^ { 0 } f ( x ) d x }$ is

A) $- \pi ^ { 2 }$
B) $- \pi$
C) $\pi$
D) $\pi ^ { 2 }$
E)Impossible to determine

Question 31

In a regular partition of the interval [2, 8] into 10 subintervals, then the length of each subintervals Δx is
​

A)1/5
B)3/5
C)4/5
D)1
E)6/5

Question 32

In a regular partition of the interval [-4, 12] into 8 subintervals, then the length of each subinterval Δx is
​

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

Question 33

Compute this definite integral using geometric methods: $\int _ { 0 } ^ { 4 } \sqrt { 4 - ( x - 2 ) ^ { 2 } } d x$ ?

A) $\pi$
B) $2 \pi$
C) $4 \pi$
D) $6 \pi$
E) $8 \pi$

Question 34

Compute this definite integral using geometric methods: $\int _ { - 7 } ^ { - 4 } \sqrt { 9 - ( x + 4 ) ^ { 2 } } d x$ .

A) $9 \pi / 4$
B) $9 \pi / 2$
C) $9 \pi$
D) $- 9 \pi / 2$
E) $- 9 \pi / 4$

Question 35

Compute this definite integral using geometric methods: $\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x$ .

A) $25 \pi / 2$
B) $25 \pi$
C) $- 25 \pi / 2$
D) $- 25 \pi$
E) $50 \pi$

Question 36

A sufficient condition for a function ƒ on [a,b] to be integrable is
​

A)ƒ is non-negative.
B)ƒ is positive.
C)ƒ is bounded above.
D)ƒ is bounded below.
E)ƒ is continuous.

Question 37

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

A)The upper and lower sums of ƒ on [a,b] are equal.
B)ƒ is non-negative.
C)ƒ is positive.
D)ƒ is continuous.
E) $\int _ { a } ^ { b } f ( x ) d x$ is bounded.

Question 38

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

A) $\int _ { a } ^ { b } f ( x ) d x = f ( x ) ( b - a )$
B) $\int _ { a } ^ { b } f ( x ) d x = \int _ { a / 2 } ^ { b / 2 } 2 f ( x ) d x$
C) $\int _ { a } ^ { b } f ( x ) d x = \frac { 1 } { 2 } \int _ { 2 a } ^ { 2 b } f ( x ) d x$
D) $\int _ { a } ^ { b } f ( x ) d x > 0$
E)is bounded. $\int _ { a } ^ { b } f ( x ) d x$

Question 39

Let ƒ be an integrable function on [a, b]. Which of the following is always true?

A) $\int _ { a } ^ { b } f ( x ) d x > b - a$
B)
$\int _ { a } ^ { b } f ( x ) d x < 2 \int _ { a } ^ { b } f ( x ) d x$
C)
$\int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$
D)
$\left[ \int _ { a } ^ { b } f ( x ) d x \right] ^ { 2 } > \int _ { a } ^ { b } f ( x ) d x$
E)
$\int _ { a } ^ { b } f ( x ) d x \leq \left| \int _ { a } ^ { b } f ( x ) d x \right|$

Question 40

Let ƒ be an integrable function on [a,b], and $s _ { n }$ and $s _ { n }$ be the lower and upper sum of a partition of [a,b], respectively. Which of the following is always true?

A) $s _ { n } = S _ { n }$
B) $\lim _ { n \rightarrow \infty } s _ { n } = \lim _ { n \rightarrow \infty } S _ { n }$
C) $\int _ { a } ^ { b } f ( x ) d x \neq 0$
D)ƒ is continuous on [a,b].
E) $\int _ { a } ^ { b } f ( x ) d x \geq 0$

Question 41

Let ƒ be an integrable function on [a,b]. Which of the following is always true?

A) $\int _ { a } ^ { b } f ( x ) d x < \int _ { a } ^ { b } f ( b ) d x$
B) $\int _ { a } ^ { b } f ( x ) d x > \int _ { a } ^ { b } f ( a ) d x$
C) $\int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$
D)is a real number. $\int _ { a } ^ { b } f ( x ) d x$
E) $\int _ { a } ^ { b } f ( x ) d x < a f ( b )$

Question 42

Let ƒ be an integrable function on [a,b]. Which of the following is not always true?

A) $\int _ { b } ^ { b } f ( x ) d x = 0$
B)ƒ is a continuous function on [a,b].
C) $\int _ { a } ^ { b } f ( x ) d x$ is a real number.
D) $\left| \int _ { a } ^ { b } f ( x ) d x \right| \geq 0$
E) $\int _ { a } ^ { b } f ( x ) d x = - \int _ { b } ^ { a } f ( x ) d x$

Question 43

Let ƒ be an integrable function on [a,b]. Which of the following is not always true?

A) $\int _ { a } ^ { a } f ( x ) d x = \int _ { b } ^ { b } f ( x ) d x$
B) $\int _ { a } ^ { b } k d x = k ( b - a )$
C) $\int _ { a } ^ { b } f ( x ) d x$ is bounded.
D) $\int _ { a } ^ { b } k d x = - k ( a - b )$
E) $2 \int _ { a } ^ { b } f ( x ) d x > \frac { 1 } { 2 } \int _ { a } ^ { b } f ( x ) d x$

Question 44

The derivative $\frac { d } { d x } \left[ \int _ { 2 } ^ { x } \frac { \sqrt { t } } { t ^ { 2 } + 1 } d t \right]$ is

A) $\frac { x ^ { 2 } + 1 } { \sqrt { x } }$
B) $\frac { \sqrt { x } } { x ^ { 2 } + 1 }$
C) $\frac { \sqrt { t } } { t ^ { 2 } + 1 }$
D) $\frac { t ^ { 2 } + 1 } { \sqrt { t } }$
E) $\frac { 2 } { 9 \sqrt { x } }$

Question 45

The derivative $\frac { d } { d x } \left[ \int _ { 1 } ^ { x } \ln t d t \right]$ is

A) $\frac { 1 } { x }$
B) $\frac { 1 } { \ln x }$
C) $\ln x$
D) $\ln t$
E) $\frac { 1 } { t }$

Question 46

The derivative $\frac { d } { d x } \left[ \int _ { 0 } ^ { x } e ^ { t ^ { 2 } } d t \right]$ is

A) $e ^ { x ^ { 2 } } - 1$
B) $e ^ { x ^ { 2 } }$
C) $\frac { e ^ { x ^ { 2 } } } { 2 x }$
D) $2 x e ^ { x ^ { 2 } }$
E) $2 x e ^ { x ^ { 2 } } - 1$

Question 47

The derivative $\frac { d } { d x } \left[ \int _ { 0 } ^ { x } \cos ^ { 2 } t d t \right]$ is

A) $\cos ^ { 2 } x$
B) $\cos ^ { 2 } x - 1$
C) $\sin ^ { 2 } x - 1$
D) $\sin ^ { 2 } x$
E) $\frac { \cos ^ { 3 } x } { 3 } - 1$

Question 48

The derivative $\frac { d } { d x } \left[ \int _ { x } ^ { 3 } 2 ^ { t } d t \right]$ is

A) $2 ^ { x + 2 } - 8$
B) $8 - 2 ^ { x + 1 }$
C) $- 2 ^ { x + 1 }$
D) $8 - \frac { 2 ^ { x + 1 } } { x + 1 }$
E) $- 2 ^ { x }$

Question 49

The derivative $\frac { d } { d x } \left[ \int _ { x } ^ { 1 } \sin ^ { - 1 } t d t \right]$ is

A) $- \sin ^ { - 1 } x$
B) $- \sin ^ { - 1 } x - \frac { \pi } { 2 }$
C) $- \sin ^ { - 1 } x + \frac { \pi } { 2 }$
D) $\sin ^ { - 1 } x - \frac { \pi } { 2 }$
E) $\sin ^ { - 1 } x + \frac { \pi } { 2 }$

Question 50

The derivative $\frac { d } { d x } \left[ \int _ { 3 } ^ { x ^ { 2 } + 1 } 2 ^ { t } d t \right]$ is

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

Question 51

The derivative $\frac { d } { d x } \left[ \int _ { 0 } ^ { \sin x } \sqrt { 1 - t ^ { 2 } } d t \right]$ is

A)cos x
B)cos² x
C)sec x
D)sec² x
E)csc² x

Question 52

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } } ^ { 0 } 5 ^ { t } d t \right]$ is

A) $x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
B) $- x ^ { 2 } \left( 5 ^ { x ^ { 2 } } \right)$
C) $2 x \left( 5 ^ { x ^ { 2 } } \right)$
D) $- 2 x \left( 5 ^ { x ^ { 2 } } \right)$
E) $- 2 x \left( 5 ^ { x ^ { 2 } } \right) + 1$

Question 53

The derivative $\frac { d } { d x } \left[ \int _ { x ^ { 2 } + 1 } ^ { 4 } \ln \left( \frac { 1 } { t } \right) d t \right]$ is

A) $2 x \ln \left( x ^ { 2 } + 1 \right)$
B) $2 x \ln \left( \frac { 1 } { x ^ { 2 } + 1 } \right)$
C) $2 x \ln \left( x ^ { 2 } + 1 \right) - \ln \left( \frac { 1 } { 4 } \right)$
D) $2 x \ln \left( \frac { 1 } { x ^ { 2 } + 1 } \right) - \ln \left( \frac { 1 } { 4 } \right)$
E) $2 x \ln \left( x ^ { 2 } + 1 \right) - \ln 4$

Question 54

If $F ( x ) = \int _ { x } ^ { - 3 } ( 1 - 2 t ) d t$ , what is F(0)?

A)-12
B)12
C)9
D)-6
E)6

Question 55

If $F ( x ) = \int _ { 1 } ^ { x } ( \sqrt { t } + 1 ) d t$ , what is F(4)?

A)1
B) $\frac { 31 } { 3 }$
C)12
D) $\frac { 23 } { 3 }$
E) $\frac { 28 } { 3 }$

Question 56

By part 2 of the Fundamental Theorem of Calculus, $\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \sec ^ { 2 } x d x$ is

A) $1 - \frac { 1 } { \sqrt { 3 } }$
B) $1 - \sqrt { 3 }$
C) $\sqrt { 3 } - 1$
D) $\frac { 1 } { \sqrt { 3 } } - 1$
E) $\frac { 1 } { 2 }$

Question 57

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { 1 } e ^ { x } d x$ is

A) $e ^ { 2 } - 1$
B) $e ^ { 2 }$
C) $e + 1$
D) $e$
E) $e - 1$

Question 58

By part 2 of the Fundamental Theorem of Calculus, $\int _ { - 2 } ^ { 2 } d x$ is

A)0
B)1
C)2
D)3
E)4

Question 59

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { \pi } { 2 } } ( 2 x + \cos x ) d x$ is

A) $\frac { \pi ^ { 2 } + 4 } { 4 }$
B) $\frac { \pi ^ { 2 } - 4 } { 4 }$
C) $\frac { \pi ^ { 2 } + 2 } { 4 }$
D) $\frac { \pi ^ { 2 } - 2 } { 4 }$
E) $\pi ^ { 2 }$

Question 60

By part 2 of the Fundamental Theorem of Calculus, $\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$ is

A) $\frac { \pi } { 6 }$
B) $\frac { \pi } { 4 }$
C) $\frac { \pi } { 2 }$
D) $\pi$
E) $\frac { 7 \pi } { 6 }$

Question 61

By part 2 of the Fundamental Theorem of Calculus, $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sec x \tan x d x$ is

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

Question 62

Let A denote the area enclosed by the graph $f ( x ) = 3 x ^ { 2 },$ the x-axis, and the lines $x = 1$ and $x = 3$ . By part 2 of the Fundamental Theorem of Calculus, A is

A)20
B)22
C)24
D)26
E)28

Question 63

Let A denote the area enclosed by the graph $f ( x ) = \cos x,$ the x-axis, and the lines $x = 0$ and $x = \frac { \pi } { 2 }$ . By part 2 of the Fundamental Theorem of Calculus, A is

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

Question 64

Let A denote the area enclosed by the graph $f ( x ) = \sqrt { x },$ the x-axis, and the lines $x = 1$ and $x = 9$ . By part 2 of the Fundamental Theorem of Calculus, A is

A) $\frac { 50 } { 3 }$
B) $\frac { 52 } { 3 }$
C) $\frac { 54 } { 3 }$
D) $\frac { 56 } { 3 }$
E) $\frac { 58 } { 3 }$

Question 65

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { x }$ , the x-axis, and the lines x = 1 and x = e. By part 2 of the Fundamental Theorem of Calculus, A is

A) $1 - \frac { 1 } { \mathrm { e } }$
B) $1 - \frac { 1 } { \mathrm { e } ^ { 2 } }$
C) $\frac { 1 } { 2 }$
D)1
E) $\frac { 1 } { \mathrm { e } ^ { 2 } }$

Question 66

The rate of water consumption (in hundreds of gallons per year) in an office building since its opening in 1995 is modeled by the function $w = \frac { 1 } { 4 } t$ , where tis the number of years after 1995. Which integral represents the total number of gallons consumed between 1996 and 2001?

A) $\int _ { 0 } ^ { 6 } \frac { 1 } { 4 } t d t$
B) $\int _ { 0 } ^ { 5 } \frac { 1 } { 4 } t d t$
C) $\int _ { 1 } ^ { 6 } \frac { 1 } { 4 } t d t$
D) $\int _ { 1 } ^ { 6 } \frac { 1 } { 4 } d t$
E) $\int _ { 0 } ^ { 5 } \frac { 1 } { 4 } d t$

Question 67

Following a massive rainstorm, water flows into a storm water drainage pond for 6 hours. If V(t) denotes the volume of water in the pond t minutes after the start of flow into the pond, which integral represents the net change of water entering the pond between 2 and 3 hours?

A) $\int _ { 2 } ^ { 3 } V ^ { \prime } ( t ) d t$
B) $\int _ { 0 } ^ { 1 } V ^ { \prime } ( t ) d t$
C) $\int _ { 120 } ^ { 180 } V ^ { \prime } ( t ) d t$
D) $\int _ { 2 } ^ { 3 } V ( t ) d t$
E) $\int _ { 120 } ^ { 180 } V ( t ) d t$

Question 68

Let $\int _ { 1 } ^ { 5 } f ( x ) d x = 4$ and $\int _ { 1 } ^ { 2 } f ( x ) d x = 2$ Then $\int _ { 2 } ^ { 5 } f ( x ) d x$ is

A)-2
B)2
C)4
D)6
E)8

Question 69

Let $\int _ { - 2 } ^ { 4 } f ( x ) d x = 12$ and $\int _ { - 2 } ^ { 0 } f ( x ) d x = 7$ Then $\int _ { 0 } ^ { 4 } f ( x ) d x$ is

A)-7
B)-5
C)5
D)7
E)19

Question 70

Let $\int _ { 6 } ^ { 10 } f ( x ) d x = 15$ and $\int _ { 4 } ^ { 6 } f ( x ) d x = - 6$ Then $\int _ { 10 } ^ { 4 } f ( x ) d x$ is

A) $- 21$
B) $- 9$
C)9
D)10
E)21

Question 71

Let $\int _ { 1 } ^ { 3 } f ( x ) d x = 14$ and $\int _ { 8 } ^ { 3 } f ( x ) d x = - 5$ Then $\int _ { 1 } ^ { 8 } f ( x ) d x$ is

A) $- 19$
B) $- 9$
C)9
D)14
E)19

Question 72

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

A)3
B)6
C)11
D)15
E)31

Question 73

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

A)-60
B)-38
C)38
D)49
E)60

Question 74

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

A) $- \frac { 49 } { 11 }$
B) $- \frac { 49 } { 6 }$
C) $\frac { 49 } { 6 }$
D) $\frac { 49 } { 11 }$
E)7

Question 75

Let $\int _ { - 1 } ^ { 1 } f ( x ) d x = 4$ , and $\int _ { 1 } ^ { 4 } f ( x ) d x = 3,$ then $\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,$ is

A)1
B)2
C)3
D)4
E)5

Question 76

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sin ^ { 2 } x d x$ are

A) $\frac { 1 } { 2 } , 1$
B) $\frac { 3 } { 4 } , 1$
C) $\frac { 1 } { 2 } , \frac { 3 } { 4 }$
D) $0 , \frac { 1 } { 2 }$
E) $0 , \frac { 3 } { 4 }$

Question 77

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { e } \ln x d x$ are

A) $0,1$
B) $0 , e$
C) $1 , e$
D) $\frac { 1 } { \mathrm { e } } , 1$
E) $0 , \frac { 1 } { \mathrm { e } }$

Question 78

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x$ are

A) $1,2$
B) $1,4$
C) $2,4$
D) $\frac { 1 } { 2 } , 1$
E) $\frac { 1 } { 4 } , 1$

Question 79

The bounds m and M used in the Bounds on an Integral Theorem for $\int _ { - 1 } ^ { 2 } \left[ 2 - ( x - 1 ) ^ { 2 } \right] d x$ are

A) $- 1,2$
B) $- 2,1$
C) $- 2 , - 1$
D) $- 2,2$
E) $- 2,3$

Question 80

Let $f ( x ) = x ^ { 3 }$ If $c \in ( - 1,1 )$ such that $\frac { \int _ { - 1 } ^ { 1 } x ^ { 3 } d x } { 2 } = f ( c ),$ then c is

A) $- \frac { 1 } { 4 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 1 } { 4 }$
D) $\frac { 1 } { 2 }$
E)0