Solved: Laplace Transforms Let be a function defined for all positive values of The

In Exercises 14, decide whether the integral is improper. Explain your reasoning.

In Exercises 14, decide whether the integral is improper. Explain your reasoning.

In Exercises 14, decide whether the integral is improper. Explain your reasoning.

In Exercises 14, decide whether the integral is improper. Explain your reasoning.

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. dx 4 0 1 x dx

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 4 3 1 x  332 dx 4

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. dx 2 0 1 x  12 dx

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 2 0 1 x  123 dx 2

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. dx 0 ex dx x

In Exercises 510, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 0  e2x dx 0

Writing In Exercises 1114, explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. 1 1 1 x2 dx  2

Writing In Exercises 1114, explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. 2 2 2 x  13 dx  8 9

Writing In Exercises 1114, explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer. sec x dx  0 0 ex dx  0

Writing In Exercises 1114, explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer.  0 sec x dx  0

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 1 1 x2 dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1 5 x3 dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 1 3 3 x dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1 4 4 x dx 1

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 0  xe2x dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0 xex2 dx 0

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 0 x2ex dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0 x  1ex dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. sin bx dx, 0 ex cos x dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. a > 0 0 eax sin bx dx,

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 4 1 xln x3 dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1 ln x x dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx  2 4 x2 dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0 x3 x2 12 dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 0 1 ex ex dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0 ex 1 ex dx 0

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. dx 0 cos x dx

In Exercises 1532, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0 sin x 2 dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 1 0 1 x2 dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 4 0 8 x dx 1

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 8 0 1 3 8  x dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 6 0 4 6  x dx 8

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 1 0 x ln x dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. e 0 ln x2 dx 1

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. sec  d 2 0 tan  d

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 2 0 sec  d

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 4 2 2 xx2  4 dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 2 0 1 4  x2 dx 4

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 4 2 1 x2  4 dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 2 0 1 4  x2 dx 4

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 2 0 1 3 x  1 dx

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 3 1 2 x  283 dx 2

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. dx 0 4 xx 6 dx 3

In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 1 1 x ln x dx

In Exercises 49 and 50, determine all values of p for which the improper integral converges. dx 1 1 xp dx W

In Exercises 49 and 50, determine all values of p for which the improper integral converges. 1 0 1 xp dx

Use mathematical induction to verify that the following integral converges for any positive integer

Given continuous functions and such that on the interval prove the following. (a) If converges, then converges. (b) If diverges, then diverges.

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. dx 1 0 1 x3 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 1 0 1 3 x dx 1

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. dx 1 1 x3 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 0 x4ex dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 1 1 x2 5 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 2 1 x  1 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 2 1 3 xx  1 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 1 1 xx 1 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 0 ex 2 dx

In Exercises 5362, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 2 1 x ln x dx

Describe the different types of improper integrals.

Define the terms and when working with improper integrals.

Explain why

Consider the integral To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

Area In Exercises 6770, find the area of the unbounded shaded region.

Area In Exercises 6770, find the area of the unbounded shaded region.

Area In Exercises 6770, find the area of the unbounded shaded region.

Area In Exercises 6770, find the area of the unbounded shaded region.

Area and Volume In Exercises 71 and 72, consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the -axis. (c) Find the volume of the solid generated by revolving the region about the -axis. y e y 0, x 0

Area and Volume In Exercises 71 and 72, consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the -axis. (c) Find the volume of the solid generated by revolving the region about the -axis. y y 0, x 1 1

Arc Length Sketch the graph of the hypocycloid of four cusps and find its perimeter.

Arc Length Find the arc length of the graph of over the interval

Surface Area The region bounded by is revolved about the axis to form a torus. Find the surface area of the torus.

Surface Area Find the area of the surface formed by revolving the graph of on the interval about the -axis.

Propulsion In Exercises 77 and 78, use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earths surface? (b) How far has the rocket traveled when half the total work has occurred?

Propulsion In Exercises 77 and 78, use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earths surface? (b) How far has the rocket traveled when half the total work has occurred?

Probability A nonnegative function is called a probability density function if The probability that lies between and is given by The expected value of is given by In Exercises 79 and 80, (a) show that the nonnegative function is a probability density function, (b) find and (c) find ft  1 7et7, 0, t 0 t < 0

Probability A nonnegative function is called a probability density function if The probability that lies between and is given by The expected value of is given by In Exercises 79 and 80, (a) show that the nonnegative function is a probability density function, (b) find and (c) find ft  2 5e2t5, 0, t 0 t < 0

Capitalized Cost In Exercises 81 and 82, find the capitalized cost of an asset (a) for years, (b) for years, and (c) forever. The capitalized cost is given by

Capitalized Cost In Exercises 81 and 82, find the capitalized cost of an asset (a) for years, (b) for years, and (c) forever. The capitalized cost is given by

Electromagnetic Theory The magnetic potential at a point on the axis of a circular coil is given by where and are constants. Find

Gravitational Force A semi-infinite uniform rod occupies the nonnegative -axis. The rod has a linear density which means that a segment of length has a mass of A particle of mass is located at the point The gravitational force that the rod exerts on the mass is given by where is the gravitational constant. Find

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is continuous on and then converges.

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is continuous on and diverges, then

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is continuous on and then

True or False? In Exercises 8588, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of is symmetric with respect to the origin or the axis, then converges if and only if converges.

Writing (a) The improper integrals and diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function over the interval Use your knowledge of the definite integral to make an inference as to whether or not the integral converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

Exploration Consider the integral where is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for 4, 8, and 12. (c) Use the graphs to approximate the integral as (d) Use a computer algebra system to evaluate the integral for the values of in part (b). Make a conjecture about the value of the integral for any positive integer Compare your results with your answer in part (c).

The Gamma Function The Gamma Function is defined by (a) Find and (b) Use integration by parts to show that (c) Write using factorial notation where is a positive integer.

Prove that where Then evaluate each integral. (a) (b) (c)

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  1 f

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  t

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  t2

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  eat

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  cos at

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  sin at

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  cosh at

Laplace Transforms Let be a function defined for all positive values of The Laplace Transform of is defined by if the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises 93100, find the Laplace Transform of the function. ft  sinh at

Normal Probability The mean height of American men between 18 and 24 years old is 70 inches, and the standard deviation is 3 inches. An 18- to 24-year-old man is chosen at random from the population. The probability that he is 6 feet tall or taller is (Source: National Center for Health Statistics) (a) Use a graphing utility to graph the integrand. Use the graphing utility to convince yourself that the area between the axis and the integrand is 1. (b) Use a graphing utility to approximate (c) Approximate using a graphing utility. Use the graph in part (a) to explain why this result is the same as the answer in part (b).

(a) Sketch the semicircle (b) Explain why without evaluating either integral.

For what value of does the integral converge? Evaluate the integral for this value of

For what value of does the integral converge? Evaluate the integral for this value of

Volume Find the volume of the solid generated by revolving the region bounded by the graph of about the -axis.

Volume Find the volume of the solid generated by revolving the unbounded region lying between and the -axis about the -axis.

u-Substitution In Exercises 107 and 108, rewrite the improper integral as a proper integral using the given -substitution. Then use the Trapezoidal Rule with to approximate the integral. u  x 1

u-Substitution In Exercises 107 and 108, rewrite the improper integral as a proper integral using the given -substitution. Then use the Trapezoidal Rule with to approximate the integral. u  1  x 1 0 cos x 1  x dx,

(a) Use a graphing utility to graph the function (b) Show that

Let be convergent and let and be real numbers where Show that