Answer: Find the escape velocity that is needed to propel a rocket of mass out of the

Explain why each of the following integrals is improper.

Which of the following integrals are improper? Why?

Find the area under the curve from to and evaluate it for , , and . Then find the total area under this curve for .

(a) Graph the functions and in the viewing rectangles by and by . (b) Find the areas under the graphs of and from to and evaluate for , , , , , and . (c) Find the total area under each curve for , if it exists.

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx  3 1 x 2 32 dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y  0 1 s 4 1 x y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx 0  1 3 4x dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 1 2x 1 3 y dx 0

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 2 e5p dp

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.0  2r y dr 

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.x 2 s1 x 3 dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y   y 3 3y 2 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx   xex 2 d

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 esx sx y dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 0 sin2 d

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.  y cos t dt 

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y  1 1 x 2 x dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y  2 dv v 2 2v 3 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.0  ze 2z dz

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 2 ye3y y dy 0

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx  1 ln x x

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.  x 3 ex 4 y dx 

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.. y .   x 2 9 x 6 dx y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y  0 ex e 2x 3 dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx  e 1 xln x 3 d

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y  0 x arctan x 1 x 2  2 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.3 2 1 s3 x y dx 1

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.14 2 dx s 4 x 2 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.8 6 4 x 6 3 y dx 14

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y 3 2 1 x 4 dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y 1 0 dx s1 x 2 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dw 9 0 1 s 3 x 1

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y 5 0 w w 2 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.3 0 dx x 2 6x 5 y

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.csc x dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y dx 0 1 e1x x 3 dx

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.1 0 e1x x 3 y dx 0

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.y 1 0 e1x x 3 y dx 0

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent.2 0 z 2 ln z dz

5 40 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 0 ln x sx y dx

41 46 Sketch the region and find its area (if the area is finite).S x, y  x  1, 0  y  ex

41 46 Sketch the region and find its area (if the area is finite).S x, y  x  0, 0  y  e x

41 46 Sketch the region and find its area (if the area is finite).S x, y  x  1, 0  y  1x 3 x S x

41 46 Sketch the region and find its area (if the area is finite).S x, y  x  0, 0  y  xex

41 46 Sketch the region and find its area (if the area is finite).S x, y  0  x 2, 0  y  sec2 x S

41 46 Sketch the region and find its area (if the area is finite).S {x, y  2 x  0, 0  y  1sx 2 } S 

(a) If , use your calculator or computer to make a table of approximate values of for , 5, 10, 100, 1000, and 10,000. Does it appear that is convergent? (b) Use the Comparison Theorem with to show that is convergent. (c) Illustrate part (b) by graphing and on the same screen for . Use your graph to explain intui tively why is convergent.

(a) If , use your calculator or computer to make a table of approximate values of for , 10, 100, 1000, and 10,000. Does it appear that x is convergent or divergent?(b) Use the Comparison Theorem with to show that is divergent. (c) Illustrate part (b) by graphing and on the same screen for . Use your graph to explain intui tively why is divergent

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y  0 x x 3 1 dx y

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y  1 2 ex x dx x

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y  1 x 1 sx 4 x dx y

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y  0 arctan x 2 ex dx

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y 1 0 sec 2 x xsx dx

4954 Use the Comparison Theorem to determine whether the integral is convergent or divergent.y  0 sin2 x sx dx

The integral is improper for two reasons: The interval is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows:

Evaluate by the same method as in Exercise 55.

5759 Find the values of for which the integral converges and evaluate the integral for those values of .y 1 0 1 x p dx

5759 Find the values of for which the integral converges and evaluate the integral for those values of .y  e 1 x ln x p dx

5759 Find the values of for which the integral converges and evaluate the integral for those values of .y 1 0 x p ln x dx

. (a) Evaluate the integral for , , , and . (b) Guess the value of when is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

(a) Show that is divergent. (b) Show that This shows that we cant define y   f x dx lim t l  y t t fx dx lim

The average speed of molecules in an ideal gas is where is the molecular weight of the gas, is the gas constant, is the gas temperature, and is the molecular speed. Show that

We know from Example 1 that the region has infinite area. Show that by rotating about the -axis we obtain a solid with finite volume.

Use the information and data in Exercise 29 of Sec tion 6.4 to find the work required to propel a 1000-kg space vehicle out of the earths gravitational field.

Find the escape velocity that is needed to propel a rocket of mass out of the gravitational field of a planet with mass and radius . Use Newtons Law of Gravitation (see Exer cise 29 in Section 6.4) and the fact that the initial kinetic energy of supplies the needed work.

Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius the density of stars depends only on the distance from the center of the cluster. If the perceived star density is given by , where is the observed planar distance from the center of the cluster, and is the actual density, it can be shown that If the actual density of stars in a cluster is , find the perceived density .

A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let be the fraction of the companys bulbs that burn out before hours, so always lies between 0 and 1. (a) Make a rough sketch of what you think the graph of might look like. (b) What is the meaning of the derivative ? (c) What is the value of ? Why?

As we saw in Section 3.8, a radioactive substance decays exponentially: The mass at time is , where is the initial mass and is a negative constant. The mean life of an atom in the substance is M k y  0 tekt dt kFor the radioactive carbon isotope, , used in radiocarbon dating, the value of is . Find the mean life of a atom.

Determine how large the number has to be so that

Estimate the numerical value of by writing it as the sum of and . Approximate the first integral by using Simpsons Rule with and show that the second integral is smaller than , which is less than 0.0000001.

If is continuous for , the Laplace transform of is the function defined by and the domain of is the set consisting of all numbers for which the integral converges. Find the Laplace transforms of the following functions. (a) (b) (c)

Show that if for , where and are constants, then the Laplace transform exists for

Suppose that and for , where is continuous. If the Laplace transform of is and the Laplace transform of is , show that Gs sFs f 0 s  a f

If is convergent and and are real numbers, show thaty a  fx dx y  a f x dx y b  f x dx y  b fx dx 7 Review

Show that .x 0 x 2 ex 2 dx 1 2 x 0 ex 2 dx y

Show that by interpreting the integrals as areas.

Find the value of the constant for which the integral converges. Evaluate the integral for this value of .

Find the value of the constant for which the integral converges. Evaluate the integral for this value of .

. Suppose is continuous on and . Is it possible that is convergent?

Show that if and , then the following integral is convergent. x  fx dx a b y  0 x a 1 x b dx a