CalculusVolume2-SASG-03-07
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 3
Techniques of Integration
3.7 Improper Integrals
Section Exercises
Evaluate the following integrals. If the integral is not convergent, answer “divergent.”
347.
Answer: divergent
349.
Answer: 351.
Answer: 353.
Without integrating, determine whether the integral converges or diverges
by comparing the function with Answer: Converges
Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.
355.
Answer: Converges to 1/2.
357.
Answer: –4
OpenStax Calculus Volume 2
Student Answer and Solution Guide
359.
Answer: 361.
Answer: diverges
363.
Answer: diverges
365.
Answer: 1.5
367.
Answer: diverges
369.
Answer: diverges
371.
Answer: diverges
Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.
373.
compare with Answer: Both integrals diverge.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Evaluate the integrals. If the integral diverges, answer “diverges.”
375.
Answer: diverges
377.
Answer: diverges
379.
Answer: 381.
Answer: 0.0
383.
Answer: 0.0
Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.
385.
Answer: 6.0
387.
Answer: 389.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
391.
Evaluate (Be careful!) (Express your answer using three decimal places.)
Answer: 393.
Evaluate Answer: 395.
Find the area of the region bounded by the curve the x
-axis, and on the left by
Answer: 7.0
397.
Find the area under in the first quadrant.
Answer: 399.
Find the volume of the solid generated by revolving about the y
-axis the region under the curve in the first quadrant.
Answer: The Laplace transform
of a continuous function over the interval is defined by
(see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F
is the set of all real numbers s
such that the improper integral converges. Find the Laplace transform F
of each of the following functions and give the domain of F
. 401.
Answer: 403.
]
Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
OpenStax Calculus Volume 2
Student Answer and Solution Guide
405.
Use the formula for arc length to show that the circumference of the circle is
Answer: Answers will vary.
A non-negative function is a probability density function if it satisfies the following definition: The probability that a random variable x
lies between a
and b
is given by 407.
Find the probability that x
is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy. Answer: 0.8775
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter Review Exercises
For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.
409.
cannot be integrated using partial fractions.
Answer: False
411.
Integration by parts can always yield the integral.
Answer: False
For the following exercises, evaluate the integral using the specified method.
413.
using trigonometric substitution
Answer: 415.
using partial fractions
Answer: 417.
using a table of integrals or a CAS Answer: For the following exercises, integrate using whatever method you choose.
419.
Answer: 421.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.
423.
[T] Answer: M
4
=
3.312
, T
4
=
3.354
, S
4
=
3.326
425.
[T] Answer: For the following exercises, evaluate the integrals, if possible. 427.
Answer: approximately 0.2194
For the following exercises, consider the gamma function given by 429.
Extend to show that assuming is a positive integer.
Answer: This is a proof; therefore, no answer is provided.
The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
431.
[T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph. Answer: Answers may vary. Ex:
km
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Student Project
Laplace Transforms
1.
Calculate the Laplace transform of Answer: 3.
Calculate the Laplace transform of (Note, you will have to integrate by parts twice.)
Answer: 5.
Use integration by parts to evaluate (Let and Note,
by the way, that we have defined Answer: This is a proof; therefore, no answer is provided.
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