Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Calculus: Early Transcendentals

If f(x) is the slope of a trail at a distance of x miles from the start of the trail, what does 35f(x)dx represent?57E If the units for x are feet and the units for a(x) are pounds per foot, what are the units for da/dx? What units does 28a(x)dxhave?The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. v(t) = 3t 5, 0 t 3 The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. v(t) = t2 2t 3, 2 t 4 The acceleration function (in m/s2) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. a(t) = t + 4, v(0) = 5, 0 t 10 The acceleration function (in m/s2) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. a(t) = 2t + 3, v(0) = 4, 0 t 3 The linear density of a rod of length 4 m is given by (x)=9+2x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.Water flows from the bottom of a storage tank at a rate of r(t) = 200 4t liters per minute, where 0 t 50. Find the amount of water that flows from the tank during the first 10 minutes.The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t (s) v (mi/h) 0 0 10 38 20 52 30 58 40 55 50 51 60 56 70 53 80 50 90 47 100 45 Suppose that a volcano is erupting and readings of the rate r(t) at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for r(t) are tonnes (metric tons) per second. (a) Give upper and lower estimates for the total quantity Q(6) of erupted materials after six seconds. (b) Use the Midpoint Rule to estimate Q(6).The marginal cost of manufacturing x yards of a certain fabric is C(x)=30.01x+0.000006x2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.68E The graph of the acceleration a(t) of a car measured in ft/s2 is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval.Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 am by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 am to July 26th at 7:30 am. Day Inflow rate (ft3/s) July 18 5275 July 19 6401 July 20 2554 July 21 4249 July 22 3016 July 23 3821 July 24 2462 July 25 2628 July 26 3003 A bacteria population is 4000 at time t = 0 and its rate of growth is 1000 2t bacteria per hour after t hours. What is the population after one hour?72E 73E Evaluate the integral by making the given substitution. cos2xdx,u=2x Evaluate the integral by making the given substitution. xex2dx,u=x2 Evaluate the integral by making the given substitution. x2x3+1dx,u=x3+1 Evaluate the integral by making the given substitution. sin2cosd,u=sin Evaluate the integral by making the given substitution. x3x45dx,u=x45 Evaluate the integral by making the given substitution. 2t+1dt,u=2t+1 Evaluate the indefinite integral. x1x2dx Evaluate the indefinite integral. x2ex3dx Evaluate the indefinite integral. (12x)9dx Evaluate the indefinite integral. sint1+costdt Evaluate the indefinite integral. cos(t/2)dt Evaluate the indefinite integral. sec22d Evaluate the indefinite integral. dx53x Evaluate the indefinite integral. y2(4y3)2/3dy Evaluate the indefinite integral. cos3sind Evaluate the indefinite integral. e5rdr Evaluate the indefinite integral. eu(1eu)2du Evaluate the indefinite integral. sinxxdx Evaluate the indefinite integral. a+bx23ax+bx3dx Evaluate the indefinite integral. z2z3+1dz Evaluate the indefinite integral. (lnx)2xdx Evaluate the indefinite integral. sinxsin(cosx)dx Evaluate the indefinite integral. sec2tan3d Evaluate the indefinite integral. xx+2dx Evaluate the indefinite integral. ex1+exdx Evaluate the indefinite integral. dxax+b(a0)Evaluate the indefinite integral. (x2+1)(x3+3x)4dx Evaluate the indefinite integral. ecostsintdt Evaluate the indefinite integral. 5tsin(5t)dt Evaluate the indefinite integral. sec2xtan2xdx Evaluate the indefinite integral. (arctanx)2x2+1dx Evaluate the indefinite integral. xx2+4dx Evaluate the indefinite integral. cos(1+5t)dt Evaluate the indefinite integral. cos(/x)x2dx Evaluate the indefinite integral. cotxcsc2xdx Evaluate the indefinite integral. 2t2t+3dt Evaluate the indefinite integral. sinh2xcoshxdx Evaluate the indefinite integral. dtcos2t1+tant Evaluate the indefinite integral. sin2x1+cos2xdx Evaluate the indefinite integral. sinx1+cos2xdx Evaluate the indefinite integral. cotxdx Evaluate the indefinite integral. cos(lnt)tdt Evaluate the indefinite integral. dx1x2sin1x Evaluate the indefinite integral. x1+x4dx Evaluate the indefinite integral. 1+x1+x2dx Evaluate the indefinite integral. x22+xdx Evaluate the indefinite integral. x(2x+5)8dx Evaluate the indefinite integral. x3x2+1dx Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). x(x21)3dx 50E 51E Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). sinxcos4xdx 53E Evaluate the definite integral. 01(3t1)50dt Evaluate the definite integral. 011+7x3dx Evaluate the definite integral. 03dx5x+1 Evaluate the definite integral. 0/6sintcos2tdt Evaluate the definite integral. /32/3csc2(12t)dt Evaluate the definite integral. 12e1/xx2dx Evaluate the definite integral. 01xex2dx Evaluate the definite integral. /4/4(x3+x4tanx)dx Evaluate the definite integral. 0/2cosxsin(sinx)dx Evaluate the definite integral. 013dx(1+2x)23 Evaluate the definite integral. 0axa2x2dx Evaluate the definite integral. 0axx2+a2dx(a0)Evaluate the definite integral. /3/3x4sinxdx Evaluate the definite integral. 12xx1dx Evaluate the definite integral. 04x1+2xdx Evaluate the definite integral. ee4dxxlnx Evaluate the definite integral. 02(x1)e(x1)2dx Evaluate the definite integral. 01ez+1ez+zdz 72E Evaluate the definite integral. 01dx(1+x)4 Verify that f(x)=sinx3 is an odd function and use that fact to show that 023sinx3dx1 75E 76E Evaluate 22(x+3)4x2dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.Evaluate 01x1x4dx by making a substitution and interpreting the resulting integral in terms of an area.Which of the following areas are equal? Why?A model for the basal metabolism rate, in kcal/h, of a young man is R(t) = 85 0.18 cos(t/12), where t is the time in hours measured from 5:00 am. What is the total basal metabolism of this man, 024R(t)dt, over a 24-hour time period?An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 100e0.01t liters per minute. How much oil leaks out during the first hour?A bacteria population starts with 400 bacteria and grows at a rate of r(t) = (450.268)e1.12567t bacteria per hour. How many bacteria will there be after three hours?Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f(t)=12sin(2t/5) has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time t.The rate of growth of a fish population was modeled by the equation G(t)=60,000e0.6t(1+5e0.6t)2 where t is measured in years and G in kilograms per year. If the biomass was 25,000 kg in the year 2000, what is the predicted biomass for the year 2020?Dialysis treatment removes urea and other waste products from a patients blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation u(t)=rVC0ert/V where r is the rate of flow of blood through the dialyzer (in mL/min), V is the volume of the patients blood (in mL), and C0 is the amount of urea in the blood (in mg) at time t = 0. Evaluate the integral 030u(t)dt and interpret it.Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after t weeks is dxdt=5000(1100(t+10)2)calculators/week (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.If f is continuous and 04f(x)dx=10, find 02f(2x)dx.If f is continuous and 09f(x)dx=4, find 03xf(x2)dx.89E 90E If a and b are positive numbers, show that 01xa(1x)bdx=01xb(1x)adx If f is continuous on [0, ], use the substitution u = x to show that 0xf(sinx)dx=20f(sinx)dx Use Exercise 92 to evaluate the integral 0xsinx1+cos2xdx (a) If f is continuous, prove that 0/2f(cosx)dx=0/2f(sinx)dx (b) Use part (a) to evaluate 0/2cos2xdx and 0/2sin2xdx.(a) Write an expression for a Riemann sum of a function f. Explain the meaning of the notation that you use. (b) If f(x) 0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If f(x) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.(a) Write the definition of the definite integral of a continuous function from a to b. (b) What is the geometric interpretation of abf(x)dx if f(x) 0? (c) What is the geometric interpretation of abf(x)dx if f(x) takes on both positive and negative values ? Illustrate with a diagram.State the Midpoint Rule.State both parts of the Fundamental Theorem of Calculus.(a) State the Net Change Theorem. (b) If r(t) is the rate at which water flows into a reservoir, what does t1t2r(t)dt represent?Suppose a particle moves back and forth along a straight line with velocity v(t), measured in feet per second, and acceleration a(t). (a) What is the meaning of 60120v(t)dt? (b) What is the meaning of 60120v(t)dt? (c) What is the meaning of 60120a(t)dt?(a) Explain the meaning of the indefinite integral f(x)dx. (b) What is the connection between the definite integral abf(x)dx and the indefinite integral f(x)dx?Explain exactly what is meant by the statement that differentiation and integration are inverse processes.State the Substitution Rule. In practice, how do you use it?Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are continuous on [a, b], then ab[f(x)+g(x)]dx=abf(x)dx+abg(x)dx Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are continuous on [a, b], then ab[f(x)+g(x)]dx=(abf(x)dx)(abg(x)dx)Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then ab5f(x)dx=5abf(x)dx Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then abxf(x)dx=xabf(x)dx Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b] and f(x) 0, then abf(x)dx=abf(x)dx Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [1, 3], then 13f(v)dv=f(3)f(1).Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are continuous and f(x) g(x) for a x b, then abf(x)dxabg(x)dx 8RQ 9RQ 10RQ 11RQ 12RQ 13RQ 14RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then ddx(abf(x)dx)=f(x)16RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 211x4dx=38 18RQ Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents.2RE Evaluate 01(x+1x2)dx by interpreting it in terms of areas.Express limxi=1nsinxix as a definite integral on the interval [0, ] and then evaluate the integral.If 06f(x)dx=10 and 04f(x)dx=7, find 46f(x)dx.(a) Write 15(x+2x5)dx as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b) Use the Fundamental Theorem to check your answer to part (a).The figure shows the graphs of f, f, and 0xf(t)dt. Identify each graph, and explain your choices.Evaluate: (a) 01ddx(earctanx)dx (b) ddx01(earctanx)dx (c) ddx0x(earctant)dt The graph of f consists of the three line segments shown. If g(x)=0xf(t)dt, find g(4) and g(4).10RE 11RE 12RE Evaluate the integral, if it exists. 01(1x9)dx Evaluate the integral, if it exists. 01(1x)9dx Evaluate the integral, if it exists. 19u2u2udu Evaluate the integral, if it exists. 01(u4+1)2du Evaluate the integral, if it exists. 01y(y2+1)5dy Evaluate the integral, if it exists. 02y21+y3dy Evaluate the integral, if it exists. 15dt(t4)2 20RE Evaluate the integral, if it exists. 01v2cos(v3)dv Evaluate the integral, if it exists. 11sinx1+x2dx Evaluate the integral, if it exists. /4/4t4tant2+costdt Evaluate the integral, if it exists. 01ex1+e2xdx 25RE Evaluate the integral, if it exists. 110xx24dx Evaluate the integral, if it exists. x+2x2+4xdx Evaluate the integral, if it exists. csc2x1+cotxdx Evaluate the integral, if it exists. sintcostdt Evaluate the integral, if it exists. sinxcos(cosx)dx Evaluate the integral, if it exists. exxdx Evaluate the integral, if it exists. sin(lnx)xdx Evaluate the integral, if it exists. tanxln(cosx)dx Evaluate the integral, if it exists. x1x4dx Evaluate the integral, if it exists. x31+x4dx Evaluate the integral, if it exists. sinh(1+4x)dx Evaluate the integral, if it exists. sectan1+secd Evaluate the integral, if it exists. 0/4(1+tant)3sec2tdt Evaluate the integral, if it exists. 03x24dx Evaluate the integral, if it exists. 04x1dx Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). cosx1+sinxdx Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). x2x2+1dx 43RE 44RE 45RE 46RE 47RE Find the derivative of the function. g(x)=0sinx1t21+t4dt 49RE 50RE 51RE 52RE Use the properties of integrals to verify the inequality. 01x2cosxdx13 Use the properties of integrals to verify the inequality. /4/2sinxxdx22 55RE 56RE Use the Midpoint Rule with n = 6 to approximate 03sin(x3)dx.A particle moves along a line with velocity function v(t) = t2 t, where v is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0, 5].59RE A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds. t(s) v(m/s) 0 0 0.5 4.67 1.0 7.34 1.5 8.86 2.0 9.73 2.5 10.22 3.0 10.51 3.5 10.67 4.0 10.76 4.5 10.81 5.0 10.81 A population of honeybees increased at a rate of r(t) bees per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.62RE 63RE 66RE 69RE 70RE 71RE Evaluate limn1n[(1n)9+(2n)9+(3n)9++(nn)9]1P 2P If 04e(x2)4dx=k, find the value 04xe(x2)4dx.5P 6P Evaluate limx0(1/x)0x(1tan2t)1/tdt. (Assume that the integrand is defined and continuous at t = 0; see Exercise 5.3.72.]The figure shows two regions in the first quadrant: A(t) is the area under the curve y = sin(x2) from 0 to t, and B(t) is the area of the triangle with vertices O, P, and (t, 0). Find limt0+[A(t)/B(t)]. FIGURE FOR PROBLEM 8 Find the interval [a, b] for which the value of the integral ab(2+xx2)dx is a maximum.Use an integral to estimate the sum i=110000i.(a) Evaluate 0nxdx, where n is a positive integer. (b) Evaluate abxdx, where a and b are real numbers with 0 a b.A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a height r/1+2 above the surface of the liquid.15P The figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region. FIGURE FOR PROBLEM 18 Evaluate limn(1nn+1+1nn+2++1nn+n).Find the area of the shaded region.Find the area of the shaded region.Find the area of the shaded region.Find the area of the shaded region.Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = ex, y = x2 1, x = 1, x = 1 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = sin x, y = x, x = /2, x =Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = (x 2)2, y = x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = x2 4x, y = 2x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = 1/x, y = 1/x2, x = 2 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = sin x, y =2x/, x 0 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x = 1 y2, x = y2 1 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. 4x + y2 = 12, x = y Sketch the region enclosed by the given curves and find its area. y = 12 x2, y = x2 6 Sketch the region enclosed by the given curves and find its area. y = x2, y = 4x x2 Sketch the region enclosed by the given curves and find its area. y = sec2x, y = 8 cos x, /3 x /3 Sketch the region enclosed by the given curves and find its area. y = cos x, y = 2 cos x, 0 x 2 Sketch the region enclosed by the given curves and find its area. x = 2y2, x = 4 + y2 Sketch the region enclosed by the given curves and find its area. y=x1, x y = 1 Sketch the region enclosed by the given curves and find its area. y = cos x, y = 4x2 1 Sketch the region enclosed by the given curves and find its area. x = y4, y=2x, y = 0 Sketch the region enclosed by the given curves and find its area. y = tan x, y = 2 sin x, /3 x /3 Sketch the region enclosed by the given curves and find its area. y = x3, y = x Sketch the region enclosed by the given curves and find its area. y=2x3, y=18x2, 0 x 6 Sketch the region enclosed by the given curves and find its area. y = cos x, y = 1 cos x, 0 x Sketch the region enclosed by the given curves and find its area. y = x4, y = 2 |x|Sketch the region enclosed by the given curves and find its area. y = sinh x, y = ex, x = 0, x = 2 Sketch the region enclosed by the given curves and find its area. y = 1/x, y = x, y=14x, .x 0 Sketch the region enclosed by the given curves and find its area. y=14x2, y = 2x2, x + y = 3, x 0 29E Sketch the region enclosed by the given curves and find its area. y=x1+x2, y=x9x2, x 0 Sketch the region enclosed by the given curves and find its area. y=x1+x2, y=x21+x3 Sketch the region enclosed by the given curves and find its area. y=lnxx, y=(lnx)2x Use calculus to find the area of the triangle with the given vertices. (0, 0), (3, 1), (1, 2)Use calculus to find the area of the triangle with the given vertices. (2, 0), (0, 2), (1, 1)Evaluate the integral and interpret it as the area of a region. Sketch the region. 0/2sinxcos2xdx Evaluate the integral and interpret it as the area of a region. Sketch the region. 113x2xdx Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x sin(x2), y = x4, x 0 38E 39E 40E Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y=21+x4, y = x2 42E 43E 44E Sketch the region in the xy-plane defined by the inequalities x 2y2 0, 1 x |y| 0 and find its area.Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wings cross-section.If the birth rate of a population is b(t) = 2200e0.024t people per year and the death rate is d(t) = 1460e0.018t people per year, find the area between these curves for 0 1 10. What does this area represent?In Example 5, we modeled a measles pathogenesis curve by a function f. A patient infected with the measles virus who has some immunity to the virus has a pathogenesis curve that can be modeled by, for instance, g(t) = 0.9f(t). (a) If the same threshold concentration of the virus is required for infectiousness to begin as in Example 5, on what day does this occur? (b) Let P3 be the point on the graph of g where infectiousness begins. It has been shown that infectiousness ends at a point P4 on the graph of g where the line through P3, P4 has the same slope as the line through P1, P2 in Example 5(b). On what day does infectiousness end? (c) Compute the level of infectiousness for this patient.52E Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions. (a) Which car is ahead after one minute? Explain. (b) What is the meaning of the area of the shaded region? (c) Which car is ahead after two minutes? Explain. (d) Estimate the time at which the cars are again side by side.The figure shows graphs of the marginal revenue function R and the marginal cost function C for a manufacturer. [Recall from Section 4.7 that R(x) and C(x) represent the revenue and cost when x units are manufactured. Assume that R and C are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.The curve with equation y2 = x2(x + 3) is called Tschirnhausens cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.Find the area of the region bounded by the parabola y = x2, the tangent line to this parabola at (1, 1), and the x-axis.Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.(a) Find the number a such that the line x = a bisects the area under the curve y = 1/x2, l x 4. (b) Find the number b such that the line y = b bisects the area in part (a).Find the values of c such that the area of the region bounded by the parabolas y = x2 c2 and y = c2 x2 is 576.Suppose that 0 c /2. For what value of c is the area of the region enclosed by the curves y = cos x, y = cos(x c), and x = 0 equal to the area of the region enclosed by the curves y = cos(x c), x = , and y = 0?For what values of m do the line y = mx and the curve y = x/(x2 + 1) enclose a region? Find the area of the region.Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x + 1, y = 0, x = 0, x = 2; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = 1/x, y = 0, x = 1, x = 4; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y=x1, y = 0, x = 5; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = ex, y = 0, x = 1, x = 1; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. x=2y, x = 0, y = 9; about the y-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. 2x = y2, x = 0, y = 4; about the y-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x3, y = x, x 0; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = 6 x2, y = 2; about the x-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y2 = x, x = 2y; about the y-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. x = 2 y2, x = y4; about the y-axis Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x2, x = y2; about y = 1 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y= x3, y= 1, x = 2; about y= 3 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y= 1 + sec x, y = 3; about y= 1 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = sin x, y =cos x, 0 x /4; about y = 1 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x3, y = 0, x = 1; about x = 2 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. xy = 1, y = 0, x = 1 , x = 2; about x = 1 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. x = y2, x = 1 y2; about x = 3 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x, y=0, x = 2, x = 4; about x = 1 Refer to the figure and find the volume generated by rotating the given region about the specified line. R1 about OA

Page: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]