Centers of MassFind the centroid of the region bounded

Find the lengths of the curves in Exercises 17–20. \(y=x^2-(\ln x)/8,\quad\ \ \ 1\le x\le2\) Equation Transcription: Text Transcription: y = x^2 - (ln x)/8, 1 leq x leq 2

Problem 9AAE Surface Area A particle of mass m starts from rest at time t = 0 and is moved along the x-axis with constant acceleration a from x = 0 to x = h against a variable force of magnitude F(t) = t 2. Find the work done

Problem 18AAE Fluid Force A vertical rectangular plate is submerged in a fluid with its top edge parallel to the fluid’s surface. Show that the force exerted by the fluid on one side of the plate equals the average value of the pressure up and down the plate times the area of the plate.

Find the lengths of the curves in Exercises 17–20. \(x=y^{2/3},\quad\ \ \ 1\le y\le8\) Equation Transcription: Text Transcription: x = y^2/3, 1 leq y leq 8

Problem 12AAE Centers of Mass If you haul a telephone pole on a two-wheeled carriage behind a truck, you want the wheels to be 3 ft or so behind the pole’s center of mass to provide an adequate “tongue” weight. The 40-ft wooden telephone poles used by Verizon have a 27-in. circumference at the top and a 43.5-in. circumference at the base. About how far from the top is the center of mass?

Problem 12PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = ? and y = 2 about the line y = 2

Problem 13AAE Centers of Mass Suppose that a thin metal plate of area A and constant density occupies a region R in the xy-plane, and let My be the plate’s moment about the y-axis. Show that the plate’s moment about the line x = b is a. if the plate lies to the right of the line, and b. if the plate lies to the left of the line.

Problem 13PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the region bounded by the curve x = ey2 and the lines y = 0, x = 0, and y = 1 about the x-axis.

Problem 14AAE Centers of Mass Find the center of mass of a thin plate covering the region bounded by the curve y2 = 4ax and the line x = a, a = positive constant if the density at (x, y) is directly proportional to (a) x, (b)

Problem 14PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving about the x-axis the region bounded by y = 2 tan x, y = 0, x = -?/4, and x = ?/4 (The region lies in the first and third quadrants and resembles a skewed bowtie.)

Problem 15AAE Centers of Mass a. Find the centroid of the region in the first quadrant bounded by two concentric circles and the coordinate axes, if the circles have radii a and b, 0 < a < b and their centers are at the origin. b. Find the limits of the coordinates of the centroid as a approaches b and discuss the meaning of the result.

Problem 15PE Volumes Find the volumes of the solids. Volume of a solid sphere hole A round hole of radius is bored through the center of a solid sphere of a radius 2 ft. Find the volume of material removed from the sphere.

Problem 16AAE Centers of Mass A triangular corner is cut from a square 1 ft on a side. The area of the triangle removed is 36 in2 If the centroid of the remaining region is 7 in. from one side of the original square, how far is it from the remaining sides?

Problem 10QGY How do you locate the center of mass of a thin plate bounded by two curves y = ƒ(x) and y = g(x) over a ? x ? b?

Problem 11AAE Centers of Mass Find the centroid of the region bounded below by the x-axis and above by the curve y = 1 - xn, n an even positive integer. What is the limiting position of the centroid as

Problem 11PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the “triangular” region bounded by the x-axis, the line and the curve y = tan x in the first quadrant about the x-axis

Problem 16PE Volumes Find the volumes of the solids. Volume of a football The profile of a football resembles the ellipse shown here. Find the football’s volume to the nearest cubic inch.

Problem 17AAE Fluid Force A triangular plate ABC is submerged in water with its plane vertical. The side AB, 4 ft long, is 6 ft below the surface of the water, while the vertex C is 2 ft below the surface. Find the force exerted by the water on one side of the plate.

Problem 17PE Lengths of Curves Find the lengths of the curves.

Problem 8QGY What is a center of mass? a centroid?

Problem 9PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y2 + 1and on the right by the line x = 5 about (a) the x-axis; (b) the y-axis; (c) the line x = 5

Problem 20PE Lengths of Curves Find the lengths of the curves.

Problem 21PE Areas of Surfaces of Revolution Find the areas of the surfaces generated by revolving the curves about the given axes. x-axis

Problem 22PE Areas of Surfaces of Revolution Find the areas of the surfaces generated by revolving the curves about the given axes. x-axis

Problem 7QGY How do you calculate the force exerted by a liquid against a portion of a flat vertical wall? Give an example.

Problem 8AAE Surface Area At points on a circle of radius a, line segments are drawn perpendicular to the plane of the circle, the perpendicular at each point P being of length ks, where s is the length of the arc of the circle measured counterclockwise from (a, 0) to P and k is a positive constant, as shown here. Find the area of the surface formed by the perpendiculars along the arc beginning at (a, 0) and extending once around the circle.

Problem 8PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x3 and the lines x = 1 and y = 1/2about (a) the x-axis; (b) the y-axis; (c) the line x = 2; (d) the line y = 4.

Problem 1QGY How do you define and calculate the volumes of solids by the method of slicing? Give an example.

Problem 2AAE Volume and Length A solid is generated by revolving about the x-axis the region bounded by the graph of the positive continuous function y = f(x), the x-axis, and the lines x = 0 and x = a. Its volume, for all a > 0 is a2 + a. Find ƒ(x).

Problem 2PE Volumes Find the volumes of the solids. The base of the solid is the region in the first quadrant between the line y = x and the parabola The cross-sections of the solid perpendicular to the x-axis are equilateral triangles whose bases stretch from the line to the curve.

Problem 26PE Work Leaky tank truck You drove an 800-gal tank truck of water from the base of Mt. Washington to the summit and discovered on arrival that the tank was only half full. You started with a full tank, climbed at a steady rate, and accomplished the 4750-ft elevation change in 50 min. Assuming that the water leaked out at a steady rate, how much work was spent in carrying water to the top? Do not count the work done in getting yourself and the truck there. Water weighs 8 lb U.S. gal.

Problem 27PE Earth’s attraction the force of attraction on an object below Earth’s surface is directly proportional to its distance from earth’s center. Find the work done in moving a weight of w lb located a mi below Earth’s surface up tp the surface itself. Assume Earth’s radius is a constant r mi.

Problem 28PE Work Garage door spring A force of 200 N will stretch a garage door spring 0.8 m beyond its unstressed length. How far will a 300-N force stretch the spring? How much work does it take to stretch the spring this far from its unstressed length?

Problem 33PE Centers of Mass and Centroids Find the centroid of a thin, flat plate covering the region enclosed by the parabolas

Problem 34PE Centers of Mass and Centroids Find the centroid of a thin, flat plate covering the region enclosed by the x-axis, the lines x = 2 and x = -2, and the parabola y = x2.

Problem 1AAE Volume and Length A solid is generated by revolving about the x-axis the region bounded by the graph of the positive continuous function y = f(x), the x-axis, and the fixed line x = a and the variable line x = b, b > a. Its volume, for all b, is b2 - ab. Find ƒ(x).

Problem 23PE Areas of Surfaces of Revolution Find the areas of the surfaces generated by revolving the curves about the given axes. y-axis

Problem 1PE Volumes Find the volumes of the solids. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 1. The cross-sections perpendicular to the x-axis between these planes are circular disks whose diameters run from the parabola y = x2 to the parabola

Problem 24PE Areas of Surfaces of Revolution Find the areas of the surfaces generated by revolving the curves about the given axes. y-axis

Problem 25PE Work Lifting equipment A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 newton per meter. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.)

Problem 2QGY How are the disk and washer methods for calculating volumes derived from the method of slicing? Give examples of volume calculations by these methods.

Problem 3AAE Volume and Length Suppose that the increasing function ƒ(x) is smooth for x 0 and that . Let s(x) denote the length of the graph of ƒ from (0, a) to (x, ƒ(x)), x > 0. Find ƒ(x) if s(x) = Cx for some constant C. What are the allowable values for C?

Problem 3PE Volumes Find the volumes of the solids. The solid lies between planes perpendicular to the x-axis at x = ?/4 and x = 5?/4. The cross-sections between these planes are circular disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x.

Problem 29PE Work Pumping a reservoir A reservoir shaped like a right-circular cone, point down, 20 ft across the top and 8 ft deep, is full of water. How much work does it take to pump the water to a level 6 ft above the top?

Problem 30PE Work Pumping a reservoir (Continuation of Exercise 29.) The reservoir is filled to a depth of 5 ft, and the water is to be pumped to the same level as the top. How much work does it take? Reference: Exercise 29 Pumping a reservoir A reservoir shaped like a right-circular cone, point down, 20 ft across the top and 8 ft deep, is full of water. How much work does it take to pump the water to a level 6 ft above the top?

Problem 31PE Work Pumping a conical tank A right-circular conical tank, point down, with top radius 5 ft and height 10 ft is filled with a liquid whose weight-density is 60 lb/ft3. How much work does it take to pump the liquid to a point 2 ft above the tank? If the pump is driven by a motor rated at 275 ft-lb/ sec (1/ 2 hp), how long will it take to empty the tank?

Problem 9QGY How do you locate the center of mass of a thin flat plate of material? Give an example.

Problem 10AAE Surface Area Work and kinetic energy Suppose a 1.6-oz golf ball is placed on a vertical spring with force constant k = 2 lb/in. The spring is compressed 6 in. and released. About how high does the ball go (measured from the spring’s rest position)?

Problem 10PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the region bounded by the parabola y2 = 4x and the line y = x about (a) the x-axis; (b) the y-axis; (c) the line x = 4; (d) the line y = 4.

Problem 4QGY How do you find the length of the graph of a smooth function over a closed interval? Give an example. What about functions that do not have continuous first derivatives?

Problem 5AAE Volume and Length Find the volume of the solid formed by revolving the region bounded by the graphs of y = x and y = x2 about the line y = x

Problem 35PE Centers of Mass and Centroids Find the centroid of a thin, flat plate covering the “triangular” region in the first quadrant bounded by the y-axis, the parabola y = x2/4, and the line y = 4.

Problem 5PE Volumes Find the volumes of the solids. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4.The cross-sections of the solid perpendicular to the x-axis between these planes are circular disks whose diameters run from the curve x2 = 4y to the curve y2 = 4x.

Problem 36PE Centers of Mass and Centroids Find the centroid of a thin, flat plate covering the region enclosed by the parabola y2 = x and the line x = 2y

Problem 37PE Centers of Mass and Centroids Find the center of mass of a thin, flat plate covering the region enclosed by the parabola y2 = x and the line x = 2y if the density function is (Use horizontal strips.)

Problem 5QGY How do you define and calculate the area of the surface swept out by revolving the graph of a smooth function about the x-axis? Give an example. y = f(x), a ? x ? b,

Problem 6AAE Volume and Length Consider a right-circular cylinder of diameter 1. Form a wedge by making one slice parallel to the base of the cylinder completely through the cylinder, and another slice at an angle of 45° to the first slice and intersecting the first slice at the opposite edge of the cylinder (see accompanying diagram). Find the volume of the wedge.

Problem 39PE Fluid Force Trough of water The vertical triangular plate shown here is the end plate of a trough full of water (w = 62.4) .What is the fluid force against the plate?

Problem 38PE Centers of Mass and Centroids a. Find the center of mass of a thin plate of constant density covering the region between the curve and the x-axis from x = 1 to x = 9. b. Find the plate’s center of mass if, instead of being constant, the density is x. (Use vertical strips.)

Problem 6PE Volumes Find the volumes of the solids. The base of the solid is the region bounded by the parabola y2 = 4x and the line x = 1 in the xy-plane. Each cross-section perpendicular to the x-axis is an equilateral triangle with one edge in the plane. (The triangles all lie on the same side of the plane.)

Problem 40PE Fluid Force Trough of maple syrup The vertical trapezoidal plate shown here is the end plate of a trough full of maple syrup weighing 75 lb/ft3 .What is the force exerted by the syrup against the end plate of the trough when the syrup is 10 in. deep?

Problem 3QGY Describe the method of cylindrical shells. Give an example.

Problem 4AAE Volume and Length a. Show that for b. Generalize the result in part (a).

Problem 4PE Volumes Find the volumes of the solids. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 6.The cross-sections between these planes are squares whose bases run from the x-axis up to the curve

Problem 32PE Work Pumping a cylindrical tank A storage tank is a right-circular cylinder 20 ft long and 8 ft in diameter with its axis horizontal. If the tank is half full of olive oil weighing 57 lb/ft3 find the work done in emptying it through a pipe that runs from the bottom of the tank to an outlet that is 6 ft above the top of the tank.

Problem 6QGY How do you define and calculate the work done by a variable force directed along a portion of the x-axis? How do you calculate the work it takes to pump a liquid from a tank? Give examples.

Problem 7AAE Surface Area At points on the curve line segments of length are drawn perpendicular to the xy-plane. (See accompanying figure.) Find the area of the surface formed by these perpendiculars from (0, 0) to

Problem 41PE Fluid Force Force on a parabolic gate A flat vertical gate in the face of a dam is shaped like the parabolic region between the curve y = 4x2 and the line y = 4, with measurements in feet. The top of the gate lies 5 ft below the surface of the water. Find the force exerted by the water against the gate (w = 62.4) .

Problem 7PE Volumes Find the volumes of the solids. Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x4, and the lines x = 1and x = -1 about (a) the x-axis; (b) the y-axis; (c) the line x = 1; (d) the line y = 3.

Problem 42PE Fluid Force You plan to store mercury (w = 849 lb/ft3) in a vertical rectangular tank with a 1 ft square base side whose interior side wall can withstand a total fluid force of 40,000 lb. About how many cubic feet of mercury can you store in the tank at any one time?