1 Limits And Continuity 2 The Derivative 3 Topics In Differentiation 4 The Derivative In Graphing And Applications 5 Integration 6 Applications Of The Definite Integral In Geometry, Science, And Engineering 7 Principles Of Integral Evaluation 8 Mathematical Modeling With Differential Equations 9 Infinite Series 10 Parametric And Polar Curves; Conic Sections 11 Three-dimensional Space; Vectors 12 Vector-valued Functions 13 Partial Derivatives 14 Multiple Integrals 15 Topics In Vector Calculus expand_more
5.1 An Overview Of The Area Problem 5.2 The Indefinite Integral 5.3 Integration By Substitution 5.4 The Definition Of Area As A Limit; Sigma Notation 5.5 The Definite Integral 5.6 The Fundamental Theorem Of Calculus 5.7 Rectilinear Motion Revisited Using Integration 5.8 Average Value Of A Function And Its Applications 5.9 Evaluating Definite Integrals By Substitution 5.10 Logarithmic And Other Functions Defined By Integrals Chapter Questions expand_more
Problem 1QCE: Assume that g is continuous on a,b and that f is continuous on an interval containing the values of... Problem 2QCE: In each part, use the substitution to replace the given integral with an integral involving the... Problem 3QCE: Evaluate the integral by making an appropriate substitution. (a) 0sin3xdx= (b) 23xx22dx= (c)... Problem 1ES: Express the integral in terms of the variable u , but do not evaluate it. (a) 132x13dx;u=2x1 (b)... Problem 2ES: Express the integral in terms of the variable u , but do not evaluate it. (a) 1452x8dx;u=52x (b)... Problem 3ES: Express the integral in terms of the variable u , but do not evaluate it. (a) 01e2x1dx;u=2x1 (b)... Problem 4ES: Express the integral in terms of the variable u , but do not evaluate it. (a) 13tan1x1+x2dx;u=tan1x... Problem 5ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 6ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 7ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 8ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 9ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 10ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 11ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 12ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 13ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 14ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 15ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 16ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 17ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 18ES: Evaluate the definite integral two ways: first by a u- substitution in the definite integral and... Problem 19ES: Evaluate the definite integral by expressing it in terms of u and evaluating the resulting integral... Problem 20ES: Evaluate the definite integral by expressing it in terms of u and evaluating the resulting integral... Problem 21ES: Evaluate the definite integral by expressing it in terms of u and evaluating the resulting integral... Problem 22ES: Evaluate the definite integral by expressing it in terms of u and evaluating the resulting integral... Problem 23ES: A particle moves with a velocity of t=sintm/s along an s-axis . Find the distance traveled by the... Problem 24ES: A particle moves with a velocity of t=3cos2tm/s along an s-axis . Find the distance traveled by the... Problem 25ES: Find the area under the curve y=9/x+22 over the interval 1,1 . Problem 26ES Problem 27ES: Find the area of the region enclosed by the graphs of y=1/19x2,y=0,x=0, and x=16 . Problem 28ES: Find the area of the region enclosed by the graphs of y=sin1x,x=0, and y=/2 . Problem 29ES: Find the average value of fx=x/5x2+12 over the interval 0,2 . Problem 30ES: Find the average value of fx=e3x/1+e6x over the interval ln3/6,0 . Problem 31ES: Evaluate the integrals by any method. 15dx2x1 Problem 32ES: Evaluate the integrals by any method. 125x1dx Problem 33ES: Evaluate the integrals by any method. 11x2dxx3+9 Problem 34ES: Evaluate the integrals by any method. /26sinxcosx+15dx Problem 35ES: Evaluate the integrals by any method. 13x+2x2+4x+7dx Problem 36ES: Evaluate the integrals by any method. 12dxx26x+9 Problem 37ES: Evaluate the integrals by any method. 0/44sinxcosxdx Problem 38ES: Evaluate the integrals by any method. 0/4tanxsec2xdx Problem 39ES: Evaluate the integrals by any method. 05xcosx2dx Problem 40ES: Evaluate the integrals by any method. 2421xsinxdx Problem 41ES: Evaluate the integrals by any method. /12/9sec23d Problem 42ES: Evaluate the integrals by any method. 0/6tan2d Problem 43ES: Evaluate the integrals by any method. 01y2dy43y Problem 44ES: Evaluate the integrals by any method. 14xdx5+x Problem 45ES: Evaluate the integrals by any method. 0edx2x+e Problem 46ES: Evaluate the integrals by any method. 12xex2dx Problem 47ES: Evaluate the integrals by any method. 01x43x4dx Problem 48ES: Evaluate the integrals by any method. 121x4xdx Problem 49ES: Evaluate the integrals by any method. 01/311+9x2dx Problem 50ES: Evaluate the integrals by any method. 12x3+x4dx Problem 51ES: (a) Use a CAS to find the exact value of the integral 0/6sin4xcos3xdx (b) Confirm the exact value of... Problem 52ES: (a) Use a CAS to find the exact value of the integral /4/4tan4xdx (b) Confirm the exact value of... Problem 53ES: (a) Find 01f3x+1dx if 14fxdx=5 . (b) Find 03f3xdx if 09fxdx=5 . (c) Find 20xfx2dx if 04fxdx=1 . Problem 54ES: Given that m and n are positive integers, show that 01xm1xndx=01xn1xmdx by making a substitution. Do... Problem 55ES: Given that n is a positive integers, show that 0/2sinnxdx=0/2cosnxdx by using a trigonometric... Problem 56ES: Given that n is a positive integer, evaluate the integral 01x1xndx Problem 57ES: Medication can be administered to a patient in different ways. For a given method, let ct denote the... Problem 58ES: Medication can be administered to a patient in different ways. For a given method, let ct denote the... Problem 59ES: Medication can be administered to a patient in different ways. For a given method, let ct denote the... Problem 60ES: Medication can be administered to a patient in different ways. For a given method, let ct denote the... Problem 61ES: Suppose that at time t=0 there are 750 bacteria in a growth medium and the bacteria population yt... Problem 62ES: Suppose that a particle moving along a coordinate line has velocity t=25+10e0.05tft/s . (a) What is... Problem 63ES: (a) The accompanying table shows the fraction of the Moon that is illuminated (as seen from Earth)... Problem 64ES: Electricity is supplied to homes in the form of alternating current, which means that the voltage... Problem 65ES: Find a positive value of k such that the area under the graph of y=e2x over the interval 0,k is 3... Problem 66ES: Use a graphing utility to estimate the value of kk0 so that the region enclosed by y=1/1+kx2,y=0,x=0... Problem 67ES: (a) Find the limit limn+k=1nsink/nn by evaluating an appropriate definite integral over the interval... Problem 68ES: Let I=1111+x2dx (a) Explain why I0 . (b) Show that the substitution x=1/u results in I=1111+x2dx=I... Problem 69ES: (a) Prove that if f is an odd function, then aafxdx=0 (b) Prove that if f is an even function, then... Problem 70ES: Show that if f and g are continuous functions, then 0tftxgxdx=0tfxgtxdx Problem 71ES: (a) Let I=0afxfx+faxdx Show that I=a/2 . (b) Use the result of part (a) to find 03xx+3xdx (c) Use... Problem 72ES: Evaluate (a) 11xcosx2dx (b) 0sin8xcos5xdx format_list_bulleted