Definite Integrals Questions and Answers

39 Find the length of the astroid x2 y 3 1 1 1 0 2 3 2 3 1 X

17 y In sec x 0 x 4 18 x e e 0 y l 19 x 3 1 y 9 20 y 3 cosh 2x 0 x 1 21 y x Inx 1 x 2

3 8 Set up but do not evaluate an integral for the length of the curve 3 y x 0 x 2 5 y x lnx 1 x 4 7 x sin y 0 y 2 4 y e 1 x 3 6 x y y 0 y 3 8 y ln x 1 y 1 9 24 Find the exact length of the curve 9 y x2 0 x 2

First compute the gradient of the function f x y 6 x 7y Then evaluate it at the point 2 2 The gradient is Vf x y 0

Summarize the pertinent information obtained by apply Find the domain of f x Select the correct choice belov

6 7 3 32a Simplify the expression using a double angle formula or half angle formula cos sin 2 7 7 3 43 Write the expression as an algebraic expression in x sin 2 tan x 8 7 3 Use the cofunction identity to rewrite the following in terms of cotangent then simplify tan cot

Evaluate the indefinite integral 26x

Let f x x 3x 189x 6 Use the definition of a derivative or the derivative rules to find f x f x Use the definition of a derivative or the derivative rules to find f x f x On what interval is f increasing do not include the endpoints in the interval Interval on which f is increasing On what interval is f decreasing do not include the endpoints in the interval Interval on which f is decreasing On what interval is f concave down do not include the endpoints in the interval Interval of downward concavity On what interval is f concave up do not include the endpoints in the interval Interval of upward concavity

Use the sum of the first 10 terms to approximate the sum 5 of the series Round your answers to six decimal places Sn n 3 3 x Estimate the error S 10 S n 1 3

AA 3 1 Using Derivat answered x 2x 36x 192x

Calculate the Fourier series of the function f t 2 tl 0 1 2 It otherwise

5 16 Solving Basic Trigonometric Equations S equation 5 sin 0 V3 2 6 sin 0

Of the people who fished at Clearwater Park today 48 had a fishing license and 12 did not Of the people who fished at Mountain View Park today 40 had a license and 10 did not No one fished at both parks Suppose that one fisher from each park is chosen at random What is the probability that the fisher chosen from Clearwater had a license and the fisher chosen from Mountain View did not have a license Do not round your answer If necessary consult a list of formulas

What is the domain of g x y OB x y x 1 Select the correct choice below and fill in any answer boxes within your choice OA x y y Use a comma to separate answers as needed OC x y x xy Use a comma to separate answers as needed OD R and y Use a comma to separate answers as needed

2 a Find the area of the region inside r 3a cos 0 but outside r a a cos 0 where a is some positive constant b Graph the set of points whose polar coordinates satisfy 0 7 2 r 2 6 c Convert the polar equation r sin 0 6 into Cartesian coordinates and sketch it

6 Pts total Given the integral Q f 4y 2x dydx a 4 Points Rewrite the given integral REVERSING the order of integration Be sure to draw a NEAT well labelled diagram of R showing a windshield wiper corresponding to the NEW order of integration b 2 Points Perform the integration using the new order IF you are unable to perform part a complete the integration in the given order Max credit 1 point

In a trend that scientists attribute at least in part to global warming a certain floating cap of sea ice has been shrinking since 1980 The ice cap always shrinks in the summer and grows in winter Average minimum size of the ice cap in square miles can be approximated by A r In 2013 the radius of the ice cap was approximately 793 and was shrinking at a rate of approximately 4 1 mi yr How fast was the area changing at that time The area was changing at a rate of Round to the nearest integer as needed in 2013 mi yr mi yr yr mi mi yr mi yr

Evaluate the definite integral CT 2 sin re

se the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y x and y about the line y 1 olume

75 We know from Example 1 that the region x y x 1 0 y 1 x has infinite area Show that by rotating about the x axis obtain a solid called Gabriel s horn with finite volume

50 3 2 Which of the following integrals are improper Why dx a sec x dx b 5 x d 15 dx 1 S x dx 1 2 3 2 S 3 Find the area under the curve y 1 x from x 1 tox 1 and evaluate it for r 10 100 and 1000 Then find the total area under this curve for x 1 X S 1 dx S x 1 x dx 1 4 29 CHAPTER 7 Techniques of Integration e 17 1 S 1 e 19 fxe dx 21 fcos 21 dr 23 sin a da 25 f x 1x 27 Sze dz In x dx dx dx Copyright 2021 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02 200 203 Copyright 2021 Cangage Loaming All Rights Reserved May and te orgled scannod or deplicated in whole or in part Due in electric vights sense thint party cratcat may be suppressed from the ellonk and eChapte s Blitorial review has dormed that say upgeessed content does not matorially affect the overall learning experience Cengago Leaming serves the right to remove additional content at any time if absequent rights restrictions require it 12 f 14 16 18 20 22 S 24 1 x 26 o So 30 A x x S 2 1 x x S x 14x 1 x fede dx dx 747 dx 1 dx In x 2 dx 28 ye dy dx sin 0 e do dv S 4 20 3 dx dx 5 48 Determine whether the integral is convergent or divergent Evaluate integrals that are convergent 6 5 2x dx 7 Se dx 9 2x 4 dx 8 10 dx 51 S x y x1 0 y 1 x x YM 52 S x y x 0 0 y xe 53 S x y 0 x 2 0 y scc x M 54 S x y 2 x 0 0 y 1 x 2 dx x 4 MM 56 a If g x 1 x dx M55 a If g x sin x x use a calculator or computer to make a table of approximate values of f g x dx fort 2 5 10 100 1000 and 10 000 Does it appear that f g x dx is convergent b Use the Comparison Theorem with f x 1 x to show that f g x dx is convergent c Illustrate part b by graphing and g on the same screen for 1x10 Use your graph to explain intuitively why g x dx is convergent 1 use a calculator or computer to make a table of approximate values of g x dx for 5 10 100 1000 and 10 000 Docs it appear that g x dx is convergent or divergent b Use the Comparison Theorem with f x 1 x to show that f g x dx is divergent c Illustrate part b by graphing fand g on the same screen for 2 x 20 Use your graph to explain intuitively why g x dx is divergent 57 64 Use the Comparison Theorem to determine whether the

550 a sec x dx dx 1 x x c 15 3 Find the area under the curve y 1 x from x 1 to x t and evaluate it for t 10 100 and 1000 Then find the total area under this curve for x 1 CHAPTER 7 Techniques of Integration 1 11 S x 2 2 dx X S 13 f x 1 dx C x x 1 dx 12 S 14 Copyright 2021 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02 200 203 Copyright 2021 Congngo Learning All Rights Reserved May not be copied scanned or duplicatod in whole or in part Due to clectronic rights somo third party content may be suppressed from the eBook and or oChapter s Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1 So itx 16 dx x x dx Pou dx 3 X 1 4 x dx Evaluate integrals that are convergent 5 2x dx dx 7 Se x dx 1 9 S x 4 dx 6 5 dx 8 dx 1 10 S x 4 dx 51 S x y x 1 0 y 1 x x M52 S x y x 0 0 y xe x 53 S x y 0 x 2 0 y sec x 54 S x y 2 x 0 0 y 1 x 2 55 If a x sin x x use a calculator or computer to make

550 d S x x 3 Find the area under the curve y 1 x from x 1 to x t and evaluate it for t 10 100 and 1000 Then find the total area under this curve for x 1 dx c x x CHAPTER 7 Techniques of Integration 1 11 fx 23 2 dx 21 X 13 S x 1 dx 15 f x x dx x4 1 17 2 dx 19 xe xe do Copyright 2021 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02 200 203 Copyright 2021 Congngo Leaming All Rights Reserved May not be copied scanned or doplientod in whole or in part Due to clectronic rights somo third party content may be suppressed from the eBook and or oChapter 3 Editorial review has deemed that any suppressed content does not materially affect the overall leaming experienco Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it cos 2t dt 1 So it dx x 12 S 14 S4x dx 16 S 18 31 22 X x 1 x x x3 20 x 1 d dx 100 dx Case 7 S 2x dx 1 9 2x 4 dx 8 dx 10 1 x 4 dx 51 S x y x1 0 y 1 x x M52 S x y x 0 0 y xe 53 S x y 0 x 2 0 y sec x 54 S x y 2 x 0 0 y 1 x 2 55 a If g x sin x x use a calculator or computer to make a table of approximate values of f g x dx for t 2 5 10 100 1000 and 10 000 Does it appear that f g x dx is convergent b Use the Comparison Theorem with f x 1 x to show that g x dx is convergent c Illustrate part b by graphing f and g on the same screen for 1x10 Use your graph to explain intuitively why g x dx is convergent

on s Sn using the indicated value of n int 3 5 sqrt x 2 1 d x n 4 mathrm s mathrm n Simplify your answer Round the final answer to three decimal places as needed Roi

D O none of these answers Question 2 find all possible rational zer f x 2x5 3x 2x 10

Carl s house payment is 1 610 per month and his car payment is 468 per month If Carl s take home pay is 3 500 per month what percentage does Carl spend on his home and car Enter your answer as a percent rounded to 2 decimal places Loan payments to income ratio

The disposable income from your part time job in 2019 was 15 700 In 2018 you borrowed 1 090 at 10 percent interest You repaid your loan with interest in 2019 How much would you have available for spending in 2019 Do not round intermediate calculations Spending amount available in 2019

The velocity in ft sec of a particle moving along the x axis is given by the function e te What is the average velocity of the particle from time t 0 to time t 3 A B 20 086 ft sec 26 447 ft sec 32 809 ft sec 40 671 ft sec 79 342 ft sec

Find the volume of the solid of revolution generated by revolving the region bounded by the graph of f x axis to the right of x 1 about the x axis f x Use symbolic notation and fractions where needed If the volume is not defined enter DNE VIIx and the x

Find the volume of the region between the graph of f x y 36 2 and the zyplane volume

Use the meaning What does deka

Consider the curve given parametrically by r t 4 cost 3 sint 5 3 cost 4 sint 5t a Reparametrize the curve with respect to arc length starting from t 0 b Find the unit tangent vector T s c Find the curvature s

Find the exact area outside r 4 sin 20 and inside r 2

58 69 Find the area of the region under the given curve from 1 to 2 1 x 1 x x 3x x 58 y 69 y

1 point Compute each of the following sums 3 a 3k 1 k 1 5 b 3 3 k 1 5 c K k 2 1 k k 1

Evaluate the limit using L Hospital s rule ex 1 lim x 0 sin 4x

Evaluate the definite integral 9 tln t dt

Consider the improper integral a Find the value of the constant C for which the integral converges Answer C b Evaluate the integral for the value of C found in part a 8x C 1 1 2 1 42 8 dz I Answer T FA

The Laplace Transform turns a given function f f t into a new function F F s by way of an improper integral P s e f t dt Suppose f t et 2 a Find the set of real numbers s for which the improper integral defining F s converges Answer in interval notation Answer b Find a simple formula for F s valid on the interval described in part a Answer F s

a Solve for a a So RE 1 25 x dx 0 006 b Find the largest interval of values for R such that R 1 25 x da

Analyze the given improper integral If it converges provide its numerical value If it diverges enter one of inf or inf if either applies or DNE otherwise 1 log z S dz Remark When determining limits like lim zlog z we can use asymptotics If they aren t clear to you in this form you can try to the subsitution e to deal with 2 0

Use the trapezoidal rule with n 8 to approximate S cos x I dx Keep at least 2 decimal places accuracy in your final apr

1 c Find the exact value of f 0 X e 1 X e x dx

Derive the Laplace transform of the following time domain functions sinat b S dt C sint T e f dt t a sint t COST d f cost dr S T

69 72 Find the volume obtained by rotating the region bounded by the curves about the given axis 69 y sin x y 0 2 x 70 y sin x y 0 0 x T about the x axis 71 y sin x y cos x 0 x 4 about y 1 about the x axis

What is the z score with a confidence level of 95 when finding the margin of error for the mean of a normally distributed population from a sample O 0 99 1 65 1 96 2 58

A group of volunteers is responsible for distributing meals to shut ins throughout a small community The number of meals each volunteer distributes per week is shown here 16 39 30 20 39 43 32 21 28 45 11 23 Please state the median from this data Note the data is different from the previous question

A solid is constructed between the graphs of y f x and y g x and the lines I a and 2 b sa that the cross sections of the solid perpendicular to the z axis and to the zy plane are disks Top y f x Bottom y g x a each slice is a circle b Which of the following integrals represents the volume of this solid ST f r 2 9 2 dz 77 S z 9 2 da f x g x dx T f x g x dx

The region bounded by the lines x a and x b and the graphs of y f x and y g z is rotated around the x axis to form a solid Top y f x Bottom y g x Which of the following integrals represents the volume of this solid O of 27 2 x f x g x dx S z g x dz O f S z g x dr 1 f x g x dr ob

et f x x 1 x for x E R Prove that f converges uniformly to the zero function on R Hint By calculus th defined in Probelm 2 is 1 2 Then prove that f x 0 respectively 1 if x 0 respectively x 0