Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

The indefinite integral $\int \frac { 2 x } { x ^ { 2 } + 4 } d x$ is

Accepted Answer

A)  $\frac { x ^ { 2 } } { x ^ { 3 } + 4 x } + C$ 
B)  $\frac { x ^ { 2 } } { x ^ { 3 } + 4 x } + C$. 
C)  $x ^ { 2 } \ln \left| x ^ { 2 } + 4 \right| + C$ 
D)  $2 x \ln \left| x ^ { 2 } + 4 \right| + C$ 
E)  $\ln \left| x ^ { 2 } + 4 \right| + C$ 
A)  $\frac { x ^ { 2 } } { x ^ { 3 } + 4 x } + C$ 
B)  $\frac { x ^ { 2 } } { x ^ { 3 } + 4 x } + C$. 
C)  $x ^ { 2 } \ln \left| x ^ { 2 } + 4 \right| + C$ 
D)  $2 x \ln \left| x ^ { 2 } + 4 \right| + C$ 
E)  $\ln \left| x ^ { 2 } + 4 \right| + C$

Question 5

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

Accepted Answer

A)  $1 - \frac { 1 } { \sqrt { 3 } }$ 
B)  $1 - \sqrt { 3 }$ 
C)  $\sqrt { 3 } - 1$ 
D)  $\frac { 1 } { \sqrt { 3 } } - 1$ 
E)  $\frac { 1 } { 2 }$ 
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 6

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

Accepted Answer

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

Question 7

The solution to the differential equation $\frac { d y } { d x } = 5$ satisfying the boundary condition y = 4 when x = 2 is

Accepted Answer

A)  $y = 5 x - 2$ 
B)  $y = 5 x - 4$ 
C)  $y = 5 x - 6$ 
D)  $y = 5 x - 8$ 
E)  $y = 5 x - 10$ 
A)  $y = 5 x - 2$ 
B)  $y = 5 x - 4$ 
C)  $y = 5 x - 6$ 
D)  $y = 5 x - 8$ 
E)  $y = 5 x - 10$

Question 8

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

Accepted Answer

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)  $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 27 } { n } \sqrt { \frac { k - 1 } { n } }$ 
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 9

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

Accepted Answer

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

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

Suppose S₃is 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

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

The indefinite integral $\int \frac { 2 x } { x ^ { 2 } + 4 } d x$ is

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

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

The solution to the differential equation $\frac { d y } { d x } = 5$ satisfying the boundary condition y = 4 when x = 2 is

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

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

Preparing for Calculus

Limits and Continuity

The Derivative

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Parametric Equations; Polar Equations

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Vector Functions

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Exam 6: The Integral

The value of the definite integral ∫−12π2dx\int _ { - 1 } ^ { 2 } \pi ^ { 2 } d x∫−12​π2dx is

Suppose S3 is the lower sum of the area enclosed by the graph (x)=1x( x ) = \frac { 1 } { x }(x)=x1​ , 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

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

The indefinite integral ∫2xx2+4dx\int \frac { 2 x } { x ^ { 2 } + 4 } d x∫x2+42x​dx is

By part 2 of the Fundamental Theorem of Calculus, ∫π6π4sec⁡2xdx\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \sec ^ { 2 } x d x∫6π​4π​​sec2xdx is

Let A denote the area enclosed by the graph f(x)=cos⁡x,f ( x ) = \cos x,f(x)=cosx, the x-axis, and the lines x=0x = 0x=0 and x=π2x = \frac { \pi } { 2 }x=2π​ . By part 2 of the Fundamental Theorem of Calculus, A is

The solution to the differential equation dydx=5\frac { d y } { d x } = 5dxdy​=5 satisfying the boundary condition y = 4 when x = 2 is

Let A denote the area enclosed by the graph f(x)=x3,f ( x ) = \frac { \sqrt { x } } { 3 },f(x)=3x​​, the x-axis, and the lines x = 0 and x = 9. Graphing the region and using plane geometry, we can find that A is

Compute this definite integral using geometric methods: ∫−55−25−x2dx\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x∫−55​−25−x2​dx .

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

Suppose S₃is 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

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

The indefinite integral $\int \frac { 2 x } { x ^ { 2 } + 4 } d x$ is

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

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

The solution to the differential equation $\frac { d y } { d x } = 5$ satisfying the boundary condition y = 4 when x = 2 is

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

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