Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I
50EIn 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.The rates at which rain fell, in inches per hour, in two different locations t hours after the start of a storm are given by f(t) = 0.73t3 2t2 + t + 0.6 and g(t) = 0.17t2 0.5t + 1.1. Compute the area between the graphs for 0 t 2 and interpret your result in this context.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.54E55E56E57E58E59E60E61EFind 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-axis2E3E4EFind 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-axis6E7E8E9EFind 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-axisFind 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 = 1Find 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= 313EFind 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 = 115EFind 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 = 117E18ERefer to the figure and find the volume generated by rotating the given region about the specified line. R1 about OARefer to the figure and find the volume generated by rotating the given region about the specified line. R1 about OCRefer to the figure and find the volume generated by rotating the given region about the specified line. R1 about ABRefer to the figure and find the volume generated by rotating the given region about the specified line. R1 about BC23E24E25E26E27E28E29E30E31E32ESet up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. x2 + 4y2 = 4 (a) About y = 2 (b) About x = 2Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. y = x2, x2 + y2 = 1, y 0 (a) About the x-axis (b) About the y-axis35E36E39E40EEach integral represents the volume of a solid. Describe the solid. 01(y4y8)dy42EA CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 18, 58, 79, 94,106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver.44E(a) If the region shown in the figure is rotated about the x-axis to form a solid, use the Midpoint Rule with n = 4 to estimate the volume of the solid. (b) Estimate the volume if the region is rotated about the y-axis. Again use the Midpoint Rule with n = 4.Find the volume of the described solid S. A right circular cone with height h and base radius r48EFind the volume of the described solid S. A cap of a sphere with radius r and height hFind the volume of the described solid S. A frustum of a pyramid with square base of side b, square top of side a, and height h What happens if a = b? What happens if a = 0?51EFind the volume of the described solid S. A pyramid with height h and base an equilateral triangle with side a (a tetrahedron)Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cmFind the volume of the described solid S. The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares.55EFind the volume of the described solid S. The base of S is the triangular region with vertices (0, 0), (1, 0), and (0, 1). Cross-sections perpendicular to the y-axis are equilateral triangles.57E58EFind the volume of the described solid S. The base of S is the same base as in Exercise 58, but cross-sections perpendicular to the x-axis are isosceles triangles with height equal to the base.60E61E62E63E64E65EFind the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.67EA bowl is shaped like a hemisphere with diameter 30 cm. A heavy ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.69E70ESome of the pioneers of calculus, such as Kepler and Newton, were inspired by the problem of finding the volumes of wine barrels. (In fact Kepler published a book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas. (a) A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R cx2, h/2 x h/2, where c is a positive constant. Show that the radius of each end of the barrel is r = R d, where d = ch2/ 4. (b) Show that the volume enclosed by the barrel is V=13h(2R2+r225d2)72ELet S be the solid obtained by rotating the region shown in the figure about the y-axis. Explain why it is awkward to use slicing to find the volume V of S. Sketch a typical approximating shell. What are its circumference and height? Use shells to find V.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. y = cos4x, y = cos4x, /2 x /2; about x =24E25E26E27EIf the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with n = 5 to estimate the volume of the solid.29E30E31E32E33E34E37E38E39E40E41E42E43E44E45E46E47EUse cylindrical shells to find the volume of the solid. Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure. (a) Guess which ring has more wood in it. (b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius r through the center of a sphere of radius R and express the answer in terms of h.A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds2E3E4E5E6E7EA spring has a natural length of 40 cm. If a 60-N force is required to keep the spring compressed 10 cm, how much work is done during this compression? How much work is required to compress the spring to a length of 25 cm?Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm. (a) How much work is needed to stretch the spring from 35 cm to 40 cm? (b) How far beyond its natural length will a force of 30 N keep the spring stretched?10E11E12EShow how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (a) How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling half the rope to the top of the building?14EShow how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.16E17E18E19E20EShow how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. An aquarium 2m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg/ m3.)Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A spherical water tank, 24 ft in diameter, sits atop a 60 ft tower. The tank is filled by a hose attached to the bottom of the sphere. If a 1.5 horsepower pump is used to deliver water up to the tank, how long will it take to fill the tank? (One horsepower = 550 ft-lb of work per second.)A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 25 and 26 use the fact that water weighs 62.5 lb/ ft3.Suppose that for the tank in Exercise 23 the pump breaks down after 4. 7 105 J of work has been done. What is the depth of the water remaining in the tank?Solve Exercise 24 if the tank is half full of oil that has a density of 900 kg/m329E30E31E32E33E34E1E2EFind the average value of the function on the given interval. g(x) = 3 cos x, [/2, /2]4EFind the average value of the function on the given interval. f(t) = esin t cos t, [0, /2]6E7EFind the average value of the function on the given interval. h(u) = (ln u)/u, [1, 5]9E10E11E(a) Find the average value of f on the given interval. (b) Find c such that fave = f(c). (c) Sketch the graph off and a rectangle whose area is the same as the area under the graph of f. f(x)=2xex2,[0,2]13E14EFind the average value of f on [0, 8].The velocity graph of an accelerating car is shown. (a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity?17EThe velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r)=P4l(R2r2) where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 3.7. 7). Find the average velocity (with respect to r) over the interval 0 r R. Compare the average velocity with the maximum velocity.The linear density in a rod 8 m long is 12/x+1kg/m, where x is measured in meters from one end of the rod. Find the average density of the rod.(a) A cup of coffee has temperature 95C and takes 30 minutes to cool to 61C in a room with temperature 20C. Use Newtons Law of Cooling (Section 3.8) to show that the temperature of the coffee after t minutes is T(t)=20+75ekt where k 0.02. (b) What is the average temperature of the coffee during the first half hour?21E22E23EUse the diagram to show that if f is concave upward on [a, b], then favef(a+b2)25E26E(a) Draw two typical curves y = f(x) and y = g(x), where f(x) g(x) for a x b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area. (b) Explain how the situation changes if the curves have equations x = f(y) and x = g(y), where f(y) g(y) for c y d.2RCC3RCC4RCC5RCC(a) What is the average value of a function f on an interval [a, b]? (b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation?1RE2RE3RE4REFind the area of the region bounded by the given curves. y = sin(x/2), y = x2 2x6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20REEach integral represents the volume of a solid. Describe the solid. 0(2sinx)2dxEach integral represents the volume of a solid. Describe the solid. 042(6y)(4yy2)dy23RE24REThe height of a monument is 20 m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side 14x meters. Find the volume of the monument.26RE27RE28RE29REA steel tank has the shape of a circular cylinder oriented vertically with diameter 4 m and height 5 m. The tank is currently filled to a level of 3 m with cooking oil that has a density of 920 kg/m3. Compute the work required to pump the oil out through a 1-m spout at the top of the tank.31RE32RE33RE34RE(a) Find a positive continuous function f such that the area under the graph of f from 0 to t is A(t) = t3 for all t 0. (b) A solid is generated by rotating about the x-axis the region under the curve y = f(x), where f is a positive function and x . 0. The volume generated by the part of the curve from x = 0 to x = b is b2 for all b 0. Find the function f.2PThe figure shows a horizontal line y = c intersecting the curve y = 8x 27x3. Find the number c such that the areas of the shaded regions are equal. FIGURE FOR PROBLEM 3A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base. (a) Determine a way to slice the water into parallel rectangular cross-sections and then set up a definite integral for the volume of the water in the glass. (b) Determine a way to slice the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water. (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b). (d) Find the volume of the water in the glass from purely geometric considerations. (e) Suppose the glass is tilted until the water exactly covers half the base. In what direction can you slice the water into triangular cross-sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.5P6P7P8PThe figure shows a curve C with the property that, for every point P on the middle curve y = 2x2, the areas A and B are equal. Find an equation for C. FIGURE FOR PROBLEM 910P11PA cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed , then the container will induce a rotational motion in the liquid around the same axis. Eventually, the liquid will be rotating at the same angular speed as the container. The surface of the liquid will be convex, as indicated in the figure, because the centrifugal force on the liquid particles increases with the distance from the axis of the container. It can be shown that the surface of the liquid is a paraboloid of revolution generated by rotating the parabola y=h+2x22g about the y-axis, where g is the acceleration due to gravity. (a) Determine h as a function of . (b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top? (c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis. (i) Determine the angular speed of the container and the volume of the fluid. (ii) How far below the top of the tank is the liquid at the wall of the container? FIGURE FOR PROBLEM 1213P15PRewrite the expression without using the absolute-value symbol. 1. |5 23|Rewrite the expression without using the absolute-value symbol. 2. |5| |23|Rewrite the expression without using the absolute-value symbol. 3. | |4E5E6ERewrite the expression without using the absolute-value symbol. 7. |x 2| if x 28E9E10E11ERewrite the expression without using the absolute-value symbol. 12. |1 2x2|13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38EThe relationship between the Celsius and Fahrenheit temperature scales is given by C=59(F32), where C is the temperature in degrees Celsius and F is the temperature in degrees Fahrenheit. What interval on the Celsius scale corresponds to the temperature range 50 F 95?Use the relationship between C and F given in Exercise 39 to find the interval on the Fahrenheit scale corresponding to the temperature range 20 C 30. 39. The relationship between the Celsius and Fahrenheit temperature scales is given by C=59(F32), where C is the temperature in degrees Celsius and F is the temperature in degrees Fahrenheit. What interval on the Celsius scale corresponds to the temperature range 50 F 95?41E42E43E44E45E46E47E48E49E50E51E52E53ESolve the inequality. 54. |5x 2| 655E56E57E58E59E