Question 1

Evaluate the integral   dx.

Accepted Answer

A)  2 
B)  1 
C)  $\pi$ 
D)  e 
E)  diverges to $\infty$ 
A)  2 
B)  1 
C)  $\pi$ 
D)  e 
E)  diverges to $\infty$

Question 2

Evaluate

Accepted Answer

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

Question 3

Calculate the Trapezoid Rule approximations T2, T4, and T8 for I =   dx and use them to calculate the Simpson's Rule approximations S2, S4, and S8 and the Romberg approximations R1, R2, and R3 for I. Do all calculations to 9 decimal places. Then quote an approximate value for I to whatever precision you feel is justified by your calculations.

Accepted Answer

Question 4

Evaluate the integral   dx.

Accepted Answer

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

Question 5

Evaluate   .

Accepted Answer

A)      (   ) + C 
B)      (   ) +   + C 
C)      (   ) -   + C 
D)      (   ) + ln(1 -   ) + C 
E)      (   ) + 2x   (   ) + C 
A)      (   ) + C 
B)      (   ) +   + C 
C)      (   ) -   + C 
D)      (   ) + ln(1 -   ) + C 
E)      (   ) + 2x   (   ) + C

Question 6

Accepted Answer

A)      (x) + ln(   + 1) - x + C 
B)  (   + 1)   (x) - x + C 
C)    + C 
D)      + C 
E)      (x) - ln(   + 1) + x + C 
A)      (x) + ln(   + 1) - x + C 
B)  (   + 1)   (x) - x + C 
C)    + C 
D)      + C 
E)      (x) - ln(   + 1) + x + C

Question 7

Use Simpson's Rule with 4 subintervals to approximate I =   dx. Round your answer to 4 decimal places.

Accepted Answer

A)  S4 = 0.3467 
B)  S4 = 0.2874 
C)  S4 = 0.4009 
D)  S4 = 0.3128 
E)  S4 = 0.3465 
A)  S4 = 0.3467 
B)  S4 = 0.2874 
C)  S4 = 0.4009 
D)  S4 = 0.3128 
E)  S4 = 0.3465

Question 8

Let P(x) be a polynomial of degree 3, and suppose P(-2) = 1, P(0) = 3, and P(2) = 2.Find the exact value of   dx.

Accepted Answer

A)  10 
B)  9 
C)  11 
D)  12 
E)  8 
A)  10 
B)  9 
C)  11 
D)  12 
E)  8

Question 9

Evaluate, if convergent,   .

Accepted Answer

A)   $\pi$ 
B)  1 
C)  0 
D)    
E)  diverges to  $\infty$ 
A)   $\pi$ 
B)  1 
C)  0 
D)    
E)  diverges to  $\infty$

Question 10

Evaluate the improper integral     dx or show it diverges (to$\infty$ or -$\infty$).

Accepted Answer

A)  diverges to $\infty$ 
B)  converges to - sin(3) 
C)  converges to 3 - sin(3) 
D)  diverges to -$\infty$ 
E)  converges to sin(3) - 3cos(3) 
A)  diverges to $\infty$ 
B)  converges to - sin(3) 
C)  converges to 3 - sin(3) 
D)  diverges to -$\infty$ 
E)  converges to sin(3) - 3cos(3)

Question 11

Use Simpson's Rule with 4 and 8 subintervals to approximate I =   Give your answers to 5 decimal places. What are the actual errors in these approximations?

Accepted Answer

S₄ ≈ 2.00456, @#IMG-DLM& ≈ 2.00

Question 12

Integrate   dx.

Accepted Answer

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

Question 13

Integrate   dx.

Accepted Answer

A)  x cos x - sin x + C 
B)  x sin x + cos x + C 
C)  -x cos x + sin x + C 
D)  x sin x - cos x + C 
E)  -x sin x + cos x + C 
A)  x cos x - sin x + C 
B)  x sin x + cos x + C 
C)  -x cos x + sin x + C 
D)  x sin x - cos x + C 
E)  -x sin x + cos x + C

Question 14

Evaluate the Trapezoid and Midpoint Rule approximations   and   for   dx. Round your answer to 4 decimal places.

Accepted Answer

A)    = 0.9871;   = 1.006 
B)    = 0.9120;   = 0.9298 
C)    = 1.006;   = 0.9871 
D)    = 0.9298;   = 0.9120 
E)    = 0.9987;   = 1.001 
A)    = 0.9871;   = 1.006 
B)    = 0.9120;   = 0.9298 
C)    = 1.006;   = 0.9871 
D)    = 0.9298;   = 0.9120 
E)    = 0.9987;   = 1.001

Question 15

Let f(x) be a function such that - 9 $\le$   (x)$\le$ 3, for   [2, 4] and let J =   dx. Find the maximum absolute error involved in approximating the integral J using the Trapezoid Rule T10.

Accepted Answer

A)  0 
B)  0.04 
C)  0.03 
D)  0.02 
E)  0.06 
A)  0 
B)  0.04 
C)  0.03 
D)  0.02 
E)  0.06

Question 16

Suppose Trapezoid Rule and Midpoint Rule approximations T8 = 2.470 and M8 = 2.500 are known for the same integral I. Find the Simpson's Rule approximation S16 for I.

Accepted Answer

A)  S16 = 2.490 
B)  S16 = 2.480 
C)  S16 = 2.485 
D)  S16 = 2.495 
E)  S16 = 2.475 
A)  S16 = 2.490 
B)  S16 = 2.480 
C)  S16 = 2.485 
D)  S16 = 2.495 
E)  S16 = 2.475

Question 17

Evaluate, if convergent, the improper integral

Accepted Answer

A)    
B)  1 
C)  e 
D)    
E)  diverges to$\infty$ 
A)    
B)  1 
C)  e 
D)    
E)  diverges to$\infty$

Question 18

Evaluate the integral   .

Accepted Answer

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

Evaluate the integral dx.

Evaluate

Evaluate the integral dx.

Evaluate .

Use Simpson's Rule with 4 subintervals to approximate I = dx. Round your answer to 4 decimal places.

Let P(x) be a polynomial of degree 3, and suppose P(-2) = 1, P(0) = 3, and P(2) = 2.Find the exact value of dx.

Evaluate, if convergent, .

Evaluate the improper integral $Evaluate the improper integral dx or show it diverges (to \infty or - \infty ).$ $Evaluate the improper integral dx or show it diverges (to \infty or - \infty ).$ dx or show it diverges (to $\infty$ or - $\infty$ ).

Use Simpson's Rule with 4 and 8 subintervals to approximate I = Give your answers to 5 decimal places. What are the actual errors in these approximations?

Integrate dx.

Integrate dx.

Evaluate the Trapezoid and Midpoint Rule approximations and for dx. Round your answer to 4 decimal places.

Suppose Trapezoid Rule and Midpoint Rule approximations T₈ = 2.470 and M₈ = 2.500 are known for the same integral I. Find the Simpson's Rule approximation S₁₆ for I.

Evaluate, if convergent, the improper integral

Evaluate the integral .

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Exam 7: Techniques of Integration

Evaluate the integral dx.

Evaluate

Evaluate the integral dx.

Evaluate .

Use Simpson's Rule with 4 subintervals to approximate I = dx. Round your answer to 4 decimal places.

Let P(x) be a polynomial of degree 3, and suppose P(-2) = 1, P(0) = 3, and P(2) = 2.Find the exact value of dx.

Evaluate, if convergent, .

Evaluate the improper integral dx or show it diverges (to ∞\infty∞ or - ∞\infty∞ ).

Use Simpson's Rule with 4 and 8 subintervals to approximate I = Give your answers to 5 decimal places. What are the actual errors in these approximations?

Integrate dx.

Integrate dx.

Evaluate the Trapezoid and Midpoint Rule approximations and for dx. Round your answer to 4 decimal places.

Let f(x) be a function such that - 9 ≤\le≤ (x) ≤\le≤ 3, for x [2, 4] and let J = dx. Find the maximum absolute error involved in approximating the integral J using the Trapezoid Rule T10.

Suppose Trapezoid Rule and Midpoint Rule approximations T8 = 2.470 and M8 = 2.500 are known for the same integral I. Find the Simpson's Rule approximation S16 for I.

Evaluate, if convergent, the improper integral

Evaluate the integral .

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

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

Evaluate the improper integral $Evaluate the improper integral dx or show it diverges (to \infty or - \infty ).$ $Evaluate the improper integral dx or show it diverges (to \infty or - \infty ).$ dx or show it diverges (to $\infty$ or - $\infty$ ).

Suppose Trapezoid Rule and Midpoint Rule approximations T₈ = 2.470 and M₈ = 2.500 are known for the same integral I. Find the Simpson's Rule approximation S₁₆ for I.