Question 1

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

Accepted Answer

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

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

Accepted Answer

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

Question 3

The antiderivative $\int \tan ^ { 2 } x d x$ is

Accepted Answer

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

Question 4

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

Accepted Answer

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

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

Accepted Answer

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

Question 6

The indefinite integral $\int \frac { x } { \sqrt { x + 4 } } d x$ is

Accepted Answer

A)  $\frac { 3 } { 2 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
B)  $\frac { 2 } { 3 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
C)  $- \frac { 3 } { 4 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
D)  $- \frac { 2 } { 3 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
E)  $\frac { 3 } { 8 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
A)  $\frac { 3 } { 2 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
B)  $\frac { 2 } { 3 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
C)  $- \frac { 3 } { 4 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
D)  $- \frac { 2 } { 3 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$ 
E)  $\frac { 3 } { 8 } ( x + 4 ) ^ { 1 / 2 } ( x - 8 ) + C$

Question 7

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

Accepted Answer

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

Question 8

The definite integral $\int _ { - 1 } ^ { 1 } \left( 3 x ^ { 3 } - 2 x \right) d x$ is

Accepted Answer

A) 0 
B)  $- 3$ 
C) -2 
D) 1 
E) 2 
A) 0 
B)  $- 3$ 
C) -2 
D) 1 
E) 2

Question 9

The antiderivative $\int ( x - 4 ) ^ { 2 } d x$ is

Accepted Answer

A)  $\frac { x ^ { 3 } } { 3 } - 4 x ^ { 2 } + 16 x + C$ 
B)  $\frac { x ^ { 3 } } { 3 } - 8 x ^ { 2 } + 16 x + C$ 
C)  $\frac { x ^ { 3 } } { 3 } - 4 x ^ { 2 } - 16 x + C$ 
D)  $\frac { x ^ { 3 } } { 3 } + 8 x ^ { 2 } - 16 x + C$ 
E)  $\frac { x ^ { 3 } } { 3 } + 4 x ^ { 2 } + 16 x + C$ 
A)  $\frac { x ^ { 3 } } { 3 } - 4 x ^ { 2 } + 16 x + C$ 
B)  $\frac { x ^ { 3 } } { 3 } - 8 x ^ { 2 } + 16 x + C$ 
C)  $\frac { x ^ { 3 } } { 3 } - 4 x ^ { 2 } - 16 x + C$ 
D)  $\frac { x ^ { 3 } } { 3 } + 8 x ^ { 2 } - 16 x + C$ 
E)  $\frac { x ^ { 3 } } { 3 } + 4 x ^ { 2 } + 16 x + C$

Question 10

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

Accepted Answer

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

Question 11

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?

Accepted Answer

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)  $\int _ { a } ^ { b } f ( x ) d x \geq 0$ 
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 12

Let $f ( x ) = | x - 4 |$ If $c \in ( 0,2 )$ such that $\frac { \int _ { 0 } ^ { 2 } | x - 4 | d x } { 2 } = f ( c ),$ then c is

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 2 } { 3 }$ 
C) 1 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 5 } { 4 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 2 } { 3 }$ 
C) 1 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 5 } { 4 }$

Question 13

The indefinite integral $\int \frac { e ^ { x } } { e ^ { x } + 7 } d x$ is

Accepted Answer

A)  $\frac { e ^ { x } } { e ^ { x } + 7 } + C$ 
B)  $\frac { e ^ { x } + 7 } { e ^ { x } } + C$ 
C)  $\ln \left| e ^ { x } + 7 \right| + C$ 
D)  $\frac { 7 e ^ { x } } { e ^ { x } + 7 } + C$ 
E)  $\frac { 7 } { \left( e ^ { x } + 7 \right) ^ { 2 } } + C$ 
A)  $\frac { e ^ { x } } { e ^ { x } + 7 } + C$ 
B)  $\frac { e ^ { x } + 7 } { e ^ { x } } + C$ 
C)  $\ln \left| e ^ { x } + 7 \right| + C$ 
D)  $\frac { 7 e ^ { x } } { e ^ { x } + 7 } + C$ 
E)  $\frac { 7 } { \left( e ^ { x } + 7 \right) ^ { 2 } } + C$

Question 14

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

Accepted Answer

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

Question 15

Let $f ( x ) = 3 x ^ { 2 } + 1$ If $c \in ( 0,2 )$  such that $\frac { \int _ { 0 } ^ { 2 } \left( 3 x ^ { 2 } + 1 \right) d x } { 2 } = f ( c ),$ then c is

Accepted Answer

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

Question 16

The average value of $f ( x ) = \cos x$ on $\left[ 0 , \frac { \pi } { 6 } \right]$ is

Accepted Answer

A)  $\frac { 2 } { \pi }$ 
B)  $\frac { 3 } { \pi }$ 
C)  $\frac { 4 } { \pi }$ 
D)  $\frac { 5 } { \pi }$ 
E)  $\frac { 6 } { \pi }$ 
A)  $\frac { 2 } { \pi }$ 
B)  $\frac { 3 } { \pi }$ 
C)  $\frac { 4 } { \pi }$ 
D)  $\frac { 5 } { \pi }$ 
E)  $\frac { 6 } { \pi }$

Question 17

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

Accepted Answer

A) $\int _ { a } ^ { b } f ( x ) d x \int _ { a } ^ { b } f ( a ) d x$ 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) $\int _ { a } ^ { b } f ( x ) d x < a f ( b )$ A) $\int _ { a } ^ { b } f ( x ) d x \int _ { a } ^ { b } f ( a ) d x$ 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 18

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

Accepted Answer

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. 
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 19

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

Accepted Answer

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)  $\sin ^ { - 1 } x + \frac { \pi } { 2 }$ 
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 20

The indefinite integral $\int 5 x ^ { 2 } \sqrt [ 3 ] { x ^ { 3 } - 8 } d x$ is

Accepted Answer

A)  $\frac { 10 x } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { 10 x } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 2 / 3 } + C$ 
C)  $\frac { 5 } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$ 
D)  $\frac { 1 } { 5 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$ 
E)  $\frac { 5 } { 4 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$ 
A)  $\frac { 10 x } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 3 / 2 } + C$ 
B)  $\frac { 10 x } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 2 / 3 } + C$ 
C)  $\frac { 5 } { 3 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$ 
D)  $\frac { 1 } { 5 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$ 
E)  $\frac { 5 } { 4 } \left( x ^ { 3 } - 8 \right) ^ { 4 / 3 } + C$

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

Suppose S₄is 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

The antiderivative $\int \tan ^ { 2 } x d x$ is

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

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

The indefinite integral $\int \frac { x } { \sqrt { x + 4 } } d x$ is

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

The definite integral $\int _ { - 1 } ^ { 1 } \left( 3 x ^ { 3 } - 2 x \right) d x$ is

The antiderivative $\int ( x - 4 ) ^ { 2 } d x$ is

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

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?

Let $f ( x ) = | x - 4 |$ If $c \in ( 0,2 )$ such that $\frac { \int _ { 0 } ^ { 2 } | x - 4 | d x } { 2 } = f ( c ),$ then c is

The indefinite integral $\int \frac { e ^ { x } } { e ^ { x } + 7 } d x$ is

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

Let $f ( x ) = 3 x ^ { 2 } + 1$ If $c \in ( 0,2 )$ such that $\frac { \int _ { 0 } ^ { 2 } \left( 3 x ^ { 2 } + 1 \right) d x } { 2 } = f ( c ),$ then c is

The average value of $f ( x ) = \cos x$ on $\left[ 0 , \frac { \pi } { 6 } \right]$ is

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

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

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

The indefinite integral $\int 5 x ^ { 2 } \sqrt [ 3 ] { x ^ { 3 } - 8 } d x$ is

Preparing for Calculus

Limits and Continuity

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Exam 6: The Integral

The bounds m and M used in the Bounds on an Integral Theorem for ∫π4π3sin⁡2xdx\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \sin ^ { 2 } x d x∫4π​3π​​sin2xdx are

Suppose S4 is the lower sum of the area enclosed by the graph f(x)=x2+x,f ( x ) = x ^ { 2 } + x,f(x)=x2+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

The antiderivative ∫tan⁡2xdx\int \tan ^ { 2 } x d x∫tan2xdx is

Let ∫−11f(x)dx=4\int _ { - 1 } ^ { 1 } f ( x ) d x = 4∫−11​f(x)dx=4 , and ∫14f(x)dx=3,\int _ { 1 } ^ { 4 } f ( x ) d x = 3,∫14​f(x)dx=3, then ∫−11g(x)dx=−6,\int _ { - 1 } ^ { 1 } g ( x ) d x = - 6,∫−11​g(x)dx=−6, is

Let A denote the area enclosed by the graph f(x)=3+4−x2,f ( x ) = 3 + \sqrt { 4 - x ^ { 2 } },f(x)=3+4−x2​, the lines y = 3, x = 0, and x = 2. Graphing the region and using plane geometry, we can find that A is

The indefinite integral ∫xx+4dx\int \frac { x } { \sqrt { x + 4 } } d x∫x+4​x​dx is

Let ∫610f(x)dx=15\int _ { 6 } ^ { 10 } f ( x ) d x = 15∫610​f(x)dx=15 and ∫46f(x)dx=−6\int _ { 4 } ^ { 6 } f ( x ) d x = - 6∫46​f(x)dx=−6 Then ∫104f(x)dx\int _ { 10 } ^ { 4 } f ( x ) d x∫104​f(x)dx is

The definite integral ∫−11(3x3−2x)dx\int _ { - 1 } ^ { 1 } \left( 3 x ^ { 3 } - 2 x \right) d x∫−11​(3x3−2x)dx is

The antiderivative ∫(x−4)2dx\int ( x - 4 ) ^ { 2 } d x∫(x−4)2dx is

Let A denote the area enclosed by the graph f(x)=3x2,f ( x ) = 3 x ^ { 2 },f(x)=3x2, the x-axis, and the lines x=1x = 1x=1 and x=3x = 3x=3 . By part 2 of the Fundamental Theorem of Calculus, A is

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

Let f(x)=∣x−4∣f ( x ) = | x - 4 |f(x)=∣x−4∣ If c∈(0,2)c \in ( 0,2 )c∈(0,2) such that ∫02∣x−4∣dx2=f(c),\frac { \int _ { 0 } ^ { 2 } | x - 4 | d x } { 2 } = f ( c ),2∫02​∣x−4∣dx​=f(c), then c is

The indefinite integral ∫exex+7dx\int \frac { e ^ { x } } { e ^ { x } + 7 } d x∫ex+7ex​dx is

Suppose S4 is the lower sum of the area enclosed by the graph f(x)=2x2,f ( x ) = 2 x ^ { 2 },f(x)=2x2, 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

Let f(x)=3x2+1f ( x ) = 3 x ^ { 2 } + 1f(x)=3x2+1 If c∈(0,2)c \in ( 0,2 )c∈(0,2) such that ∫02(3x2+1)dx2=f(c),\frac { \int _ { 0 } ^ { 2 } \left( 3 x ^ { 2 } + 1 \right) d x } { 2 } = f ( c ),2∫02​(3x2+1)dx​=f(c), then c is

The average value of f(x)=cos⁡xf ( x ) = \cos xf(x)=cosx on [0,π6]\left[ 0 , \frac { \pi } { 6 } \right][0,6π​] is

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

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

The derivative ddx[∫x1sin⁡−1tdt]\frac { d } { d x } \left[ \int _ { x } ^ { 1 } \sin ^ { - 1 } t d t \right]dxd​[∫x1​sin−1tdt] is

The indefinite integral ∫5x2x3−83dx\int 5 x ^ { 2 } \sqrt [ 3 ] { x ^ { 3 } - 8 } d x∫5x23x3−8​dx is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

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

Suppose S₄is 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

The antiderivative $\int \tan ^ { 2 } x d x$ is

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

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

The indefinite integral $\int \frac { x } { \sqrt { x + 4 } } d x$ is

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

The definite integral $\int _ { - 1 } ^ { 1 } \left( 3 x ^ { 3 } - 2 x \right) d x$ is

The antiderivative $\int ( x - 4 ) ^ { 2 } d x$ is

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

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?

Let $f ( x ) = | x - 4 |$ If $c \in ( 0,2 )$ such that $\frac { \int _ { 0 } ^ { 2 } | x - 4 | d x } { 2 } = f ( c ),$ then c is

The indefinite integral $\int \frac { e ^ { x } } { e ^ { x } + 7 } d x$ is

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

Let $f ( x ) = 3 x ^ { 2 } + 1$ If $c \in ( 0,2 )$ such that $\frac { \int _ { 0 } ^ { 2 } \left( 3 x ^ { 2 } + 1 \right) d x } { 2 } = f ( c ),$ then c is

The average value of $f ( x ) = \cos x$ on $\left[ 0 , \frac { \pi } { 6 } \right]$ is

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

The indefinite integral $\int 5 x ^ { 2 } \sqrt [ 3 ] { x ^ { 3 } - 8 } d x$ is