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108EBuffon's Needle Experiment A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is P=20/2sind where is the acute angle between the needle and any one of the parallel lines. Find this probability.110E111E112EAnalyzing a Function Show that the function f(x)=01/x1t2+1dt+0x1t2+1dt is constant for x 0.114E115ECONCEPT CHECK Analyzing the Integrand Without integrating, explain why 22x(x2+1)2dx=0Finding u and du In Exercises 14, complete the table by identifying u and du for the integral. f(g(x))g(x)dxu=g(x)du=g(x)dx (8x2+1)2(16x)dx ______ ______2E3E4EFinding an Indefinite Integral In Exercises 9-30, find the indefinite integral and check the result by differentiation. (1+6x)4(6)dx6E7E8E9E10EFinding an Indefinite Integral In Exercises 526, find the indefinite integral and check the result by differentiation. x2(x31)4dx12E13E14E15E16E17E18E19EFinding an Indefinite Integral In Exercises 9-30, find the indefinite integral and check the result by differentiation. 6x2(4x39)3dx21E22E23E24EFinding an Indefinite Integral In Exercises 9-30, find the indefinite integral and check the result by differentiation. 12xdx26E27E28E29EDifferential Equation In Exercises 2730, solve the differential equation. dydx=x4x28x+1Slope Field In Exercises 35 and 36, a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation, (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point (To print an enlarged copy of the graph, go to MathGraphs.com.)(b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point. dydx=x4x2,(2,2)32E63EDifferential Equation In Exercises 37 and 38, the graph of a function f is shown. Use the differential equation and the given point to find an equation of the function. dydx=48(3x+5)333E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49EChange of Variables In Exercises 53-60, find the indefinite integral by making a change of variables. (x+1)2xdx51E52E53E54E55E56E57E58E59E60E61EEvaluating a Definite Integral In Exercises 5562, evaluate the definite integral. Use a graphing utility to verify your result. 15x2x1dx65EFinding the Area of a Region In Exercises 69-72, find the area of the region. Use a graphing utility to verify your result. 26x2x+23dx67E68E69E70E72EEven and Odd Functions In Exercises 73-76, evaluate the integral using the properties of even and odd functions as an aid. /2/2sin2xcosxdx73E74E75E76E77E79E80E81E82ESales The sales S (in thousands of units) of a seasonal product are given by the model S=74.50+43.75sint6 where t is the time in months, with t = 1 corresponding to January. Find the average sales for each time period. (a) The first quarter (0t3) (b) The second quarter (3t6) (c) The entire year (0t12)84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102EUsing the Trapezoidal Rule and Simpson's Rule In Exercises 110, use the Trapezoidal Rule and Simpsons Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. 02x2dx,n=4Using the Trapezoidal Rule and Simpson's Rule In Exercises 110, use the Trapezoidal Rule and Simpsons Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. 12(x24+1)dx,n=4Using the Trapezoidal Rule and Simpson's Rule In Exercises 110, use the Trapezoidal Rule and Simpsons Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. 02x3dx,n=44E5EUsing the Trapezoidal Rule and Simpson's Rule In Exercises 110, use the Trapezoidal Rule and Simpsons Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. 08x3dx,n=87E8E9E10E11E12E13E14E15E16E17EUsing the Trapezoidal Rule and Simpson's Rule In Exercises 1120, approximate the definite integral using the Trapezoidal Rule and Simpsons Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. 0/21+sin2xdxUsing the Trapezoidal Rule and Simpson's Rule In Exercises 1120, approximate the definite integral using the Trapezoidal Rule and Simpsons Rule with n = 4. Compare these results with the approximation of the integral using a graphing utility. 0/4xtanxdx20E21E22EEstimating Errors In Exercises 2326, use the error formulas in Theorem 4.20 to estimate the errors in approximating the integral, with n = 4, using (a) the Trapezoidal Rule and (b) Simpsons Rule. 132x3dxEstimating Errors In Exercises 2326, use the error formulas in Theorem 4.20 to estimate the errors in approximating the integral, with n = 4, using (a) the Trapezoidal Rule and (b) Simpsons Rule. 35(5x+2)dx25E26EEstimating Errors In Exercises 2730, use the error formulas in Theorem 4.20 to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 131xdx28E29EEstimating Errors In Exercises 2730, use the error formulas in Theorem 4.20 to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 0/2sinxdxEstimating Errors Using Technology In Exercises 3134, use a computer algebra system and the error formulas to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 021+xdxEstimating Errors Using Technology In Exercises 3134, use a computer algebra system and the error formulas to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 02(x+1)2/3dxEstimating Errors Using Technology In Exercises 3134, use a computer algebra system and the error formulas to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 01tanx2dxEstimating Errors Using Technology In Exercises 3134, use a computer algebra system and the error formulas to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) the Trapezoidal Rule and (b) Simpsons Rule. 01sinx2dxFinding the Area of a Region Approximate the area of the shaded region using the Trapezoidal Rule with n = 4. Simpsons Rule with n = 4.Finding the Area of a Region Approximate the area of the shaded region using the Trapezoidal Rule with n = 8. Simpsons Rule with n = 8.Area Use Simpsons Rule with n = 14 to approximate the area of the region bounded by the graphs of y=xcosx,y=0,x=0,andx=/2.Circumference The elliptic integral 830/2123sin2d gives the circumference of an ellipse. Use Simpsons Rule with n = 8 to approximate the circumference.Surveying Use the Trapezoidal Rule to estimate the number of square meters of land, where x and y are measured in meters, as shown in the figure. The land is bounded by a stream and two straight roads that meet at right angles. X 0 100 200 300 400 500 y 125 125 120 112 90 90 X 600 700 800 900 1000 y 95 88 75 35 0HOW DO YOU SEE IT? The function f(x) isconcave upward on the interval [0, 2] and the function g(x) is concave downward on the interval [0, 2]. Using the Trapezoidal Rule with n 4, which integral would be overestimated? Which integral would be underestimated? Explain your reasoning. Which rule would you use for more accurate approximations of 02f(x)dxand02g(x)dx, the Trapezoidal Rule or Simpsons Rule? Explain your reasoning.Work To determine the size of the motor required to operate a press, a company must know the amount of work done when the press moves an object linearly 5 feet. The variable force to move the object is F(x)=100x125x3 where F is given in pounds and x gives the position of the unit in feet. Use Simpsons Rule with n = 12 to approximate the work W (in foot-pounds) done through one cycle when W=05F(x)dx.42EApproximation of Pi In Exercises 43 and 44, use Simpsons Rule with n = 6 to approximate using the given equation. (In Section 5.7, you will be able to evaluate the integral using inverse trigonometric functions.) =01/261x2dxApproximation of Pi In Exercises 43 and 44, use Simpsons Rule with n = 6 to approximate using the given equation. (In Section 5.7, you will be able to evaluate the integral using inverse trigonometric functions.) =0141+x2dxUsing Simpson's Rule Use Simpsons Rule with n = 10 and a computer algebra system to approximate t in the integral equation 0tsinxdx=2.46EProof Prove that you can find a polynomial p(x) = Ax2 + Bx + C that passes through any three points (x1, y1), (x2, y2) and (x3, y3), where the xi's are distinct. Henryk Sadura/Shutterstock.comFinding an Indefinite Integral In Exercises 18, find the indefinite integral. (x6)dxFinding an Indefinite Integral In Exercises 1-8, find the indefinite integral. (x4+3)dx3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14REVelocity and Acceleration A ball is thrown vertically upward from ground level with an initial velocity of 96 feet per second. Use a(t) = 32 feet per second per second as the acceleration due to gravity. (Neglect air resistance.) How long will it take the ball to rise to its maximum height? What is the maximum height? After how many seconds is the velocity of the ball one-half the initial velocity? What is the height of the ball when its velocity is one-half the initial velocity?Velocity and Acceleration The speed of a car traveling in a straight line is reduced from 45 to 30 miles per hour in a distance of 264 feet. Find the distance in which the car can be brought to rest from 30 miles per hour, assuming the same constant deceleration.Velocity and Acceleration An airplane taking off from a runway travels 3600 feet before lifting off. The airplane starts from rest, moves with constant acceleration, and makes the run in 30 seconds. With what speed does it lift off?Modeling Data The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time t is in seconds. t v1 v2 0 0 0 5 2.5 21 10 7 38 15 16 51 20 29 60 25 45 64 30 65 65 Rewrite the velocities in feet per second. Use the regression capabilities of a graphing utility to find quadratic models for the data in part (a). Approximate the distance traveled by each car during the 30 seconds. Explain the difference in the distances.19RE20REUsing Sigma Notation In Exercises 21 and 22, use sigma notation to write the sum. 13(1)+13(2)+13(3)+...+13(10)22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59REUsing the Second Fundamental Theorem of Calculus In Exercises 57 and 58, use the Second Fundamental Theorem of Calculus to find F'(x). F(x)=1x1t2dt61RE62RE63REFinding an Indefinite Integral In Exercises 59-66, find the indefinite integral. 6x33x4+2dx65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84RE85RE86. Respiratory Cycle After exercising for a few minutes, a person has a respiratory cycle for which the rate of air intake is
Find the volume, in liters, of air inhaled during one cycle by integrating the function over the interval [0, 2].
87RE88RE89RE90RE1PSParabolic Arch Archimedes showed that the area of a parabolic arch is equal to 23 the product of the base and the height (see figure). (a) Graph the parabolic arch bounded by y = 9 x2 and the x-axis. Use an appropriate integral to find the area A. (b) Find the base and height of the arch and verify Archimedes formula. (c) Prove Archimedes formula for a general parabola.3PS4PS5PSApproximation TheTwo-Point Gaussian Quadrature Approximation for f is 11f(x)f(13)+f(13). (a) Use this formula to approximate 11cosxdx. Find the error of the approximation. (b) Use this formula to approximate 1111+x2dx. (c) Prove that the Two-Point Gaussian Quadrature Approximation is exact for all polynomials of degree 3 or less.Extrema and Points of Inflection The graph of the function f consists of the three line segments joining the points (0, 0), (2,2), (6, 2), and (8, 3). The function F is defined by the integral F(x)=0xf(t)dt (a) Sketch the graph of f. (b) Complete the table. x 0 1 2 3 4 5 6 7 8 F(x) (c) Find the extrema of F on the interval [0, 8]. (d) Determine all points of inflection of F on the interval (0, 8).8PS9PS10PS11PS12PS13PS14PSVelocity and Acceleration A car travels in a straight line for 1 hour. Its velocity v in miles per hour at six-minute intervals is shown in the table. t (hours) 0 0.1 0.2 0.3 0.4 0.5 v (mi/h) 0 10 20 40 60 50 t (hours) 0.6 0.7 0.8 0.9 1.0 v (mi/h) 40 35 40 50 65 (a) Produce a reasonable graph of the velocity function v by graphing these points and connecting them with a smooth curve. (b) Find the open intervals over which the acceleration a is positive. (c) Find the average acceleration of the car (in miles per hour per hour) over the interval [0, 0.4]. (d) What does the integral 01v(t)dx. signify? Approximate this integral using the Midpoint Rule with five subintervals. (e) Approximate the acceleration at t = 0.816PS17PSSine Integral Function The sine integral function Si(x)=0xsinttdt is often used in engineering. The function f(t)=sintt is not defined at t = 0, but its limit is 1 as t? 0. So, define f(0) = 1. Then f is continuous everywhere. (a) Use a graphing utility to graph Si(x). (b) At what values of x does Si(x) have relative maxima? (c) Find the coordinates of the first inflection point where x 0. (d) Decide whether Si(x) has any horizontal asymptotes. If so, identify each.19PS20PSEvaluating a Logarithm Using Technology In Exercises 5-8, use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral 1x(1/t)dt. ln452EEvaluating a Logarithm Using Technology In Exercises 5-8, use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral 1x(1/t)dt. ln0.84EMatching In Exercises 912, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x)=lnx+1Matching In Exercises 912, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x)=lnx7E8E9ESketching the Graph In Exercises 13-18, sketch the graph of the function and state its domain. f(x)=2lnx11ESketching the Graph In Exercises 13-18, sketch the graph of the function and state its domain. f(x)=lnx13E14ESketching a Graph In Exercises 916, sketch the graph of the function and state its domain. h(x)=ln(x+2)Sketching a Graph In Exercises 916, sketch the graph of the function and state its domain. f(x)=ln(x2)+1Using Properties of Logarithms In Exercises 19 and 20, use the properties of logarithms to approximate the indicated logarithms, given that In 20.6931 and In 31.0986. (a) ln6 (b) ln23 (c) ln81 (d) ln3Using Properties of Logarithms In Exercises 19 and 20, use the properties of logarithms to approximate the indicated logarithms, given that ln 2 = 0.6931 and In 3 = 1.0968. (a). ln 0.25 (b). ln 24 (c). ln123 (d). ln172Expanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx4Expanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx521EExpanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. ln(xyz)23E24EExpanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx1xExpanding a Logarithmic Expression In Exercises 2130, use the properties of logarithms to expand the logarithmic expression. ln(3e2)27EExpanding a Logarithmic Expression In Exercises 1928, use the properties of logarithms to expand the logarithmic expression. ln1e29E30E31E32E33E