Question 1

Evaluate   dx.

Accepted Answer

A)  2   ln   + C 
B)      +   ln   + C 
C)  ln   + C 
D)    
E)    + C 
A)  2   ln   + C 
B)      +   ln   + C 
C)  ln   + C 
D)    
E)    + C C

Question 2

Evaluate the integral   dx.

Accepted Answer

A)  ln(   + 4x + 5) + C 
B)  2 ln(   + 4x + 5) + C 
C)    ln(   + 4x + 5) + C 
D)  -2 ln(   + 4x + 5) + C 
E)  -   ln(   + 4x + 5) + C 
A)  ln(   + 4x + 5) + C 
B)  2 ln(   + 4x + 5) + C 
C)    ln(   + 4x + 5) + C 
D)  -2 ln(   + 4x + 5) + C 
E)  -   ln(   + 4x + 5) + C C

Question 3

Evaluate the integral   dx.

Accepted Answer

A)      + C 
B)      + C 
C)      + C 
D)  -     + C 
E)  -     + C 
A)      + C 
B)      + C 
C)      + C 
D)  -     + C 
E)  -     + C A

Question 4

Write the following limit as a definite integral:

Accepted Answer

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

Question 5

Express the sum   +   +   +   +..... +   using sigma notation.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

If f(x) is an even function integrable on the closed interval [0 , 2a] , a > 0 , then   =2   .

Accepted Answer

A) True 
 B)False

Question 7

Evaluate the integral   dx.

Accepted Answer

A)  ln(3   ) + C 
B)  ln   + C 
C)  3ln   + C 
D)    + C 
E)    + C 
A)  ln(3   ) + C 
B)  ln   + C 
C)  3ln   + C 
D)    + C 
E)    + C

Question 8

Express   dx as a limit of lower Riemann sums corresponding to partitions of[0, 2] into equal subintervals.

Accepted Answer

A)    dx =     . 
B)    dx =     . 
C)    dx =     . 
D)    dx =     . 
E)    dx =     . 
A)    dx =     . 
B)    dx =     . 
C)    dx =     . 
D)    dx =     . 
E)    dx =     .

Question 9

Evaluate the   .

Accepted Answer

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

Question 10

Express the sum   as a polynomial function of n.

Accepted Answer

A)  3   +     +   n 
B)  3   +     -   n 
C)  3   +     + 4n 
D)  3   + 3   - n 
E)  3   -     -   n 
A)  3   +     +   n 
B)  3   +     -   n 
C)  3   +     + 4n 
D)  3   + 3   - n 
E)  3   -     -   n

Question 11

Find the area of the region lying in the first quadrant above the curve xy = 4 and below the line x + y = 5.

Accepted Answer

A)    + 4 ln(5) square units 
B)    - 8 ln(2) square units 
C)    - 8 ln(2) square units 
D)    + 8 ln(2) square units 
E)    - 4 ln(5) square units 
A)    + 4 ln(5) square units 
B)    - 8 ln(2) square units 
C)    - 8 ln(2) square units 
D)    + 8 ln(2) square units 
E)    - 4 ln(5) square units

Question 12

Evaluate   .

Accepted Answer

A)  1 +   
B)    
C)  -   
D)  1 -   
E)  -   
A)  1 +   
B)    
C)  -   
D)  1 -   
E)  -

Question 13

Find the area of the region lying between the graphs of the equations y = sin x and y = cos x from 0 to 2 $\pi$
.

Accepted Answer

A)  2   square units 
B)  4   square units 
C)  6 square units 
D)  3   square units 
E)    square units 
A)  2   square units 
B)  4   square units 
C)  6 square units 
D)  3   square units 
E)    square units

Question 14

:     is equal to

Accepted Answer

A)  2x   
B)    -   
C)  2x   
D)  2x   
E)  2x   
A)  2x   
B)    -   
C)  2x   
D)  2x   
E)  2x

Question 15

Construct and simplify a sum approximating the area above the x-axis and under the curve y =   between x = 0 and x = 3 by using n rectangles having equal widths and tops lying under or on the curve. Find the actual area as a suitable limit.

Accepted Answer

A)    , area = 9 square units 
B)    , area = 9 square units 
C)    , area = 6 square units 
D)    , area = 6 square units 
E)    , area = 9 square units 
A)    , area = 9 square units 
B)    , area = 9 square units 
C)    , area = 6 square units 
D)    , area = 6 square units 
E)    , area = 9 square units

Question 16

Evaluate the integral   cos (   ) dx.

Accepted Answer

A)    + C 
B)    + C 
C)  -   + C 
D)  -   + C 
E)    + C 
A)    + C 
B)    + C 
C)  -   + C 
D)  -   + C 
E)    + C

Question 17

Given the piecewise continuous function f(x) =   evaluate   by using the properties of definite integrals and interpreting integrals as areas.

Accepted Answer

The answer of Given the piecewise continuous function f(x) =...

Question 18

Use the limit definition of the definite integral to evaluate   dx.

Accepted Answer

A)  10 
B)  18 
C)  6 
D)  30 
E)  9 
A)  10 
B)  18 
C)  6 
D)  30 
E)  9

Question 19

Let P denote the partition of the interval [1, 2] into 8 subintervals of equal length  x = 1/8.Evaluate the upper and lower Riemann sums U(f P) and L(f,P) for the function f(x) = 1/x.Round your answers to 4 decimal places.

Accepted Answer

A)  U(f,P) = 0.7110, L(f,P) = 0.6781 
B)  U(f,P) = 0.7254, L(f,P) = 0.6629 
C)  U(f,P) = 0.7302, L(f,P) = 0.6571 
D)  U(f,P) = 0.7378, L(f,P) = 0.6510 
E)  U(f,P) = 0.7219, L(f,P) = 0.6683 
A)  U(f,P) = 0.7110, L(f,P) = 0.6781 
B)  U(f,P) = 0.7254, L(f,P) = 0.6629 
C)  U(f,P) = 0.7302, L(f,P) = 0.6571 
D)  U(f,P) = 0.7378, L(f,P) = 0.6510 
E)  U(f,P) = 0.7219, L(f,P) = 0.6683

Question 20

Find the area of the region bounded by the graphs of the equations   = 2px and   , where   .

Accepted Answer

A)      square units 
B)      square units 
C)      square units 
D)      square units 
E)  2   square units 
A)      square units 
B)      square units 
C)      square units 
D)      square units 
E)  2   square units

Evaluate dx.

Evaluate the integral dx.

Evaluate the integral dx.

Write the following limit as a definite integral:

Express the sum + + + +..... + using sigma notation.

If f(x) is an even function integrable on the closed interval [0 , 2a] , a > 0 , then =2 .

Evaluate the integral dx.

Express dx as a limit of lower Riemann sums corresponding to partitions of[0, 2] into equal subintervals.

Evaluate the .

Express the sum as a polynomial function of n.

Find the area of the region lying in the first quadrant above the curve xy = 4 and below the line x + y = 5.

Evaluate .

Find the area of the region lying between the graphs of the equations y = sin x and y = cos x from 0 to 2 $\pi$ .

: is equal to

Construct and simplify a sum approximating the area above the x-axis and under the curve y = between x = 0 and x = 3 by using n rectangles having equal widths and tops lying under or on the curve. Find the actual area as a suitable limit.

Evaluate the integral cos ( ) dx.

Given the piecewise continuous function f(x) = evaluate by using the properties of definite integrals and interpreting integrals as areas.

Use the limit definition of the definite integral to evaluate dx.

Let P denote the partition of the interval [1, 2] into 8 subintervals of equal length x = 1/8.Evaluate the upper and lower Riemann sums U(f P) and L(f,P) for the function f(x) = 1/x.Round your answers to 4 decimal places.

Find the area of the region bounded by the graphs of the equations = 2px and , where .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 6: Integration

Evaluate dx.

Evaluate the integral dx.

Evaluate the integral dx.

Write the following limit as a definite integral:

Express the sum + + + +..... + using sigma notation.

If f(x) is an even function integrable on the closed interval [0 , 2a] , a > 0 , then =2 .

Evaluate the integral dx.

Express dx as a limit of lower Riemann sums corresponding to partitions of[0, 2] into equal subintervals.

Evaluate the .

Express the sum as a polynomial function of n.

Find the area of the region lying in the first quadrant above the curve xy = 4 and below the line x + y = 5.

Evaluate .

Find the area of the region lying between the graphs of the equations y = sin x and y = cos x from 0 to 2 π\piπ .

: is equal to

Construct and simplify a sum approximating the area above the x-axis and under the curve y = between x = 0 and x = 3 by using n rectangles having equal widths and tops lying under or on the curve. Find the actual area as a suitable limit.

Evaluate the integral cos ( ) dx.

Given the piecewise continuous function f(x) = evaluate by using the properties of definite integrals and interpreting integrals as areas.

Use the limit definition of the definite integral to evaluate dx.

Let P denote the partition of the interval [1, 2] into 8 subintervals of equal length x = 1/8.Evaluate the upper and lower Riemann sums U(f P) and L(f,P) for the function f(x) = 1/x.Round your answers to 4 decimal places.

Find the area of the region bounded by the graphs of the equations = 2px and , where .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find the area of the region lying between the graphs of the equations y = sin x and y = cos x from 0 to 2 $\pi$ .