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All Textbook Solutions for Calculus (MindTap Course List)
Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 04dxx2x2Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 01rlnrdrDetermine whether each integral is convergent or divergent. Evaluate those that are convergent. 0/2cossindDetermine whether each integral is convergent or divergent. Evaluate those that are convergent. 10e1/xx3dx40E41E42E43E44E45ESketch the region and find its area if the area is finite. S={(x,y)2x0,0y1/x+2}47E48EUse the Comparison Theorem to determine whether the integral is convergent or divergent. 0xx3+1dxUse the Comparison Theorem to determine whether the integral is convergent or divergent. 11+sin2xxdxUse the Comparison Theorem to determine whether the integral is convergent or divergent. 1x+1x4xdxUse the Comparison Theorem to determine whether the integral is convergent or divergent. 0arctanx2+exdxUse the Comparison Theorem to determine whether the integral is convergent or divergent. 01sec2xxxdxUse the Comparison Theorem to determine whether the integral is convergent or divergent. 0sin2xxdx55E56E57E58EFind the values of p for which the integral converges and evaluate the integral for those values of p. 01xplnxdxa Evaluate the integral 0xnexdx for n = 0, 1, 2, and 3. bGuess the value of 0xnexdx when n is an arbitrary positive integer. cProve your guess using mathematical induction.61E62E63E64E65EAstronomers 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 R the density of stars depends only on the distance r from the center of the cluster. If the perceived star density is given by y(s), where s is the observed planar distance from the center of the cluster, and x(r) is the actual density, it can be shown that y(s)=sR2rr2s2x(r)dr If the actual density of stars in a cluster is x(r)=12(Rr)2, find the perceived density y(s).67EAs we saw in Section 6.5, a radioactive substance decays exponentially: The mass at time t is m(t)=m(0)ekt, where m0 is the initial mass and k is a negative constant. The mean life M of an atom in the substance is M=k0tektdt For the radioactive carbon isotope, C14, used in radiocarbon dating, the value of k is 0.000121. Find the mean life of a C14 atom.In a study of the spread of illicit drug use from an enthusiastic user to a population of N users, the authors model the number of expected new users by the equation =0cN(1ekt)ketdt where c, k and are positive constants. Evaluate this integral to express in terms of c, N, k, and . Source: F. Hoppensteadt et al., Threshold Analysis of a Drug Use Epidemic Model, Mathematical Biosciences 53 1981: 7987.70E71E72EIf f(t) is continuous for t0, the Laplace transform of is the function F defined by F(s)=0f(t)estdt and the domain of F is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions. a f(t)=1 b f(t)=et c f(t)=t74E75E76E77E78E79EFind the value of the constant C for which the integral 0(xx2+1C3x+1)dx converges. Evaluate the integral for this value of C.81E82E1CC2CC3CC4CC5CC6CC7CC8CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQDetermine 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. The Midpoint Rule is always more accurate than the Trapezoidal Rule.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. a Every elementary function has an elementary derivative. b Every elementary function has an elementary antiderivative.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 [0,) and 1f(x)dx is convergent, then 0f(x)dx is convergent.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 a continuous, decreasing function on [1,) and limxf(x)=0, then 1f(x)dx is convergent.12TFQ13TFQ14TFQ1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23EEvaluate the integral. excosxdx25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40EEvaluate the integral or show that it is divergent. 11(2x+1)3dx42E43E44EEvaluate the integral or show that it is divergent. 04lnxxdxEvaluate the integral or show that it is divergent. 01123xdxEvaluate the integral or show that it is divergent. 01x1xdxEvaluate the integral or show that it is divergent. 11dxx22xEvaluate the integral or show that it is divergent. dx4x2+4x+550E51E52E53E54E55E56E57E58EVerify Formula 33 in the Table of Integrals a by differentiation and b by using a trigonometric substitution.60E61E62E63EUse a the Trapezoidal Rule, b the Midpoint Rule, and c Simpsons Rule with n=10 to approximate the given integral. Round your answers to six decimal places. 14xcosxdx65E66EThe speedometer reading v on a car was observed at 1-minute intervals and recorded in the chart. Use Simpsons Rule to estimate the distance traveled by the car. tmin vmi/h 0 40 1 42 2 45 3 49 4 52 5 54 6 56 7 57 8 57 9 55 10 5668E69E70EUse the Comparison Theorem to determine whether the integral is convergent or divergent. a 12+sinxxdx b 111+x4dxFind the area of the region bounded by the hyperbola y2x2=1 and the line y=3.73E74E75E76E77E78E79E80EThree mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they are able to determine where to slice so that each gets the same amount of pizza. Where are the cuts made?Evaluate 1x7xdx The straightforward approach would be to start with partial fractions, but that would be brutal. Try a substitution.3P4P5PA man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve see the figure. a Show that if the path followed by the boat is the graph of the function y=f(x), then f(x)=dydx=L2x2x b Determine the function y=f(x).7PIf n is a positive integer, prove that 01(lnx)ndx=(1)nn!9P10PIf 0ab, find limt0{01[bx+a(1x)]tdx}1/tGraph f(x)=sin(ex) and use the graph to estimate the value of t such that tt+1f(x)dx is a maximum. Then find the exact value of t that maximizes this integral.13P14P15PA rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v=v(t) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M=M(t) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newtons Second Law that F=Mdvdtub where the force F=Mg. Thus 1Mdvdtub=Mg Let Mi be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0=M1+M2. Then, until the fuel runs out at time t=M2/b, the mass is M=M0bt. a Substitute M=M0bt into Equation 1 and solve the resulting equation for v. Use the initial condition v(0)=0 to evaluate the constant. b Determine the velocity of the rocket at time t=M2/b. This is called the burnout velocity. c Determine the height of the rocket y=y(t) at the burnout time. d Find the height of the rocket at any time t.Use the arc length formula 3 to find the length of the curve y=2x5,1x3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.2E3E4E5E6E7E8EFind the exact length of the curve. y=1+6x3/2, 0x110EFind the exact length of the curve. y=x33+14x,1x2Find the exact length of the curve. x=y48+14y2,1y2Find the exact length of the curve. x=13y(y3),1y9Find the exact length of the curve. y=ln(cosx),0x/3Find the exact length of the curve. y=ln(secx),0x/416EFind the exact length of the curve. y=14x212lnx,1x218EFind the exact length of the curve. y=ln(1x2),0x1220E21E22E23E24EUse Simpsons Rule with n=10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y=xsinx,0x2Use Simpsons Rule with n=10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y=x3,1x6Use Simpsons Rule with n=10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y=ln(1+x3),0x5Use Simpsons Rule with n=10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by a calculator. y=ex2,0x2a Graph the curve y=x4x3,0x4. b Compute the lengths of inscribed polygons with n=1,2 and 4 sides. Divide the interval into equal sub-intervals. Illustrate by sketching these polygons as in Figure 6. c Set up an integral for the length of the curve. d Use your calculator to find the length of the curve to four decimal places. Compare with the approximations in part b.Repeat Exercise 29 for the curve y=x+sinx 0x231E32E33Ea Sketch the curve y3=x2. b Use Formulas 3 and 4 to set up two integrals for the are length from 0, 0 to 1, 1. Observe that one of these is an improper integral and evaluate both of them. c Find the length of the arc of this curve from (1,1) to 8, 4.35E36E37EThe arc length function for a curve y=f(x), where f is an increasing function, is s(x)=0x3t+5dt. a If f has y-intercept 2, find an equation for f. b What point on the graph of f is 3 units along the curve from the y-intercept? State your answer rounded to 3 decimal places.For the function f(x)=14ex+ex, prove that the arc length on any interval has the same value as the area under the curve.40EA hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y=180x245 until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.42E43Ea The figure shows a telephone wire hanging between two poles at x=b and x=b. It takes the shape of a catenary with equation y=c+acosh(x/a). Find the length of the wire. b Suppose two telephone poles are 50 ft apart and the length of the wire between the poles is 51 ft. If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?45E46Ea Set up an integral for the area of the surface obtained by rotating the curve about i the x-axis and ii the y-axis. b Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. y=tanx,0x/32E3E4Ea Set up an integral for the area of the surface obtained by rotating the curve about i the x-axis and ii the y-axis. b Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. x=y+y3,0y1a Set up an integral for the area of the surface obtained by rotating the curve about i the x-axis and ii the y-axis. b Use the numerical integration capability of a calculator to evaluate the surface areas correct to four decimal places. y=tan1x,0x2Find the exact area of the surface obtained by rotating the curve about the x-axis. y=x3,0x2Find the exact area of the surface obtained by rotating the curve about the x-axis. y=5x,3x5Find the exact area of the surface obtained by rotating the curve about the x-axis. y2=x+1,0x310EFind the exact area of the surface obtained by rotating the curve about the x-axis. y=cos(12x),0x12EFind the exact area of the surface obtained by rotating the curve about the x-axis. x=13(y2+2)3/2,1y214EThe given curve is rotated about the y-axis. Find the area of the resulting surface. y=13x3/2, 0x12The given curve is rotated about the y-axis. Find the area of the resulting surface. x2/3+y2/3=1, 0y1The given curve is rotated about the y-axis. Find the area of the resulting surface. x=a2y2, 0ya/218E19E20EUse Simpsons Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator. y=xex, 0x1Use Simpsons Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator. y=xlnx, 1x223E24E25E26EIf the region ={(x,y)x1,0y1/x} is rotated about the x-axis, the volume of the resulting solid is finite see Exercise 7.8.63. Show that the surface area is infinite. The surface is shown in the figure and is known as Gabriels horn.If the infinite curve y=ex,x0, is rotated about the x-axis, find the area of the resulting surface.29EA group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve y=ax2 about the y-axis. If the dish is to have a 10-ft diameter and a maximum depth of 2 ft, find the value of a and the surface area of the dish.a The ellipse x2a2+y2b2=1ab is rotated about the x-axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid. b If the ellipse in part a is rotated about its minor axis the y-axis, the resulting ellipsoid is called an oblate spheroid. Find the surface area of this ellipsoid.Find the surface area of the torus in Exercise 5.2.63.33E34EFind the area of the surface obtained by rotating the circle x2+y2=r2 about the line y=r.a Show that the surface area of a zone of a sphere that lies between two parallel planes is S=2Rh, where R is the radius of the sphere and h is the distance between the planes. Notice that S depends only on the distance between the planes and not on their location, provided that both planes intersect the sphere. b Show that the surface area of a zone of a cylinder with radius R and height h is the same as the surface area of the zone of a sphere in part a.Show that if we rotate the curve y=ex/2+ex/2 about the x-axis, the area of the resulting surface is the same value as the enclosed volume for any interval axb.38EFormula 4 is valid only when f(x)0. Show that when f(x) is not necessarily positive, the formula for surface area becomes S=ab2f(x)1+[f(x)]2dxAn aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find a the hydrostatic pressure on the bottom of the aquarium, b the hydrostatic force on the bottom, and c the hydrostatic force on one end of the aquarium.A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820kg/m3 to a depth of 1.5 m. Find a the hydrostatic pressure on the bottom of the tank, b the hydrostatic force on the bottom, and c the hydrostatic force on one end of the tank.3EA vertical plate is submerged or partially submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.A vertical plate is submerged or partially submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.6EA vertical plate is submerged or partially submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.A vertical plate is submerged or partially submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.A vertical plate is submerged or partially submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.10E11EA milk truck carries milk with density 64.6lb/ft3 in a horizontal cylindrical tank with diameter 6 ft. a Find the force exerted by the milk on one end of the tank when the tank is full. b What if the tank is half full?A trough is filled with a liquid of density 840kg/m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Find the hydrostatic force on a the top of the cube and b one of the sides of the cube.16EA swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, find the hydrostatic force on a the shallow end, b the deep end, one of the sides, and d the bottom of the pool.18E19E20E21E22E23E24E25ESketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y=x, y=0, x=4Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y=ex, y=0, x=0, x=128E29E30E31EFind the centroid of the region bounded by the given curves. y=x3, x+y=2, y=0Find the centroid of the region bounded by the given curves. x+y=2, x=y2Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. =4Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. =636E37E38EProve that the centroid of any triangle is located at the point of intersection of the medians. Hints: Place the axes so that the vertices are a, 0, 0, b, and c, 0. Recall that a median is a fine segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point two-thirds of the way from each vertex along the median to the opposite side.Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles from Exercise 39 and using additivity of moments.Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles from Exercise 39 and using additivity of moments.A rectangle with sides a and b is divided into two parts 1 and 2 by an arc of a parabola that has its vertex at one comer of and passes through the opposite comer. Find the centroids of both 1 and 2.43E44EUse the Theorem of Pappus to find the volume of the given solid. A cone with height h and base radius rUse the Theorem of Pappus to find the volume of the given solid. The solid obtained by rotating the triangle with vertices 2, 3, 2, 5, and 5, 4 about the x-axisThe centroid of a curve can be found by a process similar to the one we used for finding the centroid of a region. If C is a curve with length L, then the centroid is (x,y) where x=(1/L)xds and y=(1/L)yds. Here we assign appropriate limits of integration, and ds is as defined in Sections 8.1 and 8.2. The centroid often doesnt lie on the curve itself. If the curve were made of wire and placed on a weightless board, the centroid would be the balance point on the board. Find the centroid of the quarter-circle y=16x2,0x4.48E49E50E