A cardboard figure has the shape shown in Figure 8.65.The region is bounded on the left

Find the mass of a rod of length 10 cm with density (x) = ex gm/cm at a distance of x cm from the left end.

A plate occupying the region 0 x 2, 0 y 3 has density = 5 gm/cm2. Set up two integrals giving the mass of the plate, one corresponding to strips in the x-direction and one corresponding to strips in the ydirection.

A rod has length 2 meters. At a distance x meters from its left end, the density of the rod is given by (x)=2+6x gm/m. (a) Write a Riemann sum approximating the total mass of the rod. (b) Find the exact mass by converting the sum into an integral.

If a rod lies along the x-axis between a and b, the moment of the rod is - b a x(x) dx, where (x) is its density in grams/meter at a position x meters. Find the moment and center of mass of the rod in Problem 3.

The density of cars (in cars per mile) down a 20-mile stretch of the Pennsylvania Turnpike is approximated by (x) = 300 2 + sin 4 x + 0.15 , at a distance x miles from the Breezewood toll plaza. (a) Sketch a graph of this function for 0 x 20. (b) Write a Riemann sum that approximates the total number of cars on this 20-mile stretch. (c) Find the total number of cars on the 20-mile stretch.

(a) Find a Riemann sum which approximates the total mass of a 3 5 rectangular sheet, whose density per unit area at a distance x from one of the sides of length 5 is 1/(1 + x4). (b) Calculate the mass.

A point mass of 2 grams located 3 centimeters to the left of the origin and a point mass of 5 grams located 4 centimeters to the right of the origin are connected by a thin, light rod. Find the center of mass of the system

Find the center of mass of a system containing three point masses of 5 gm, 3 gm, and 1 gm located respectively at x = 10, x = 1, and x = 2.

Find the mass of the block 0 x 10, 0 y 3, 0 z 1, whose density , is given by = 2 z for 0 z 1.

Problems 1012 refer to a colony of bats which flies out of a cave each night to eat insects. To estimate the colonys size, a naturalist counts samples of bats at different distances from the cave. Table 8.3 gives n, her count per hectare, at a distance r km from the cave. For instance, she counts 300 bats in one hectare at the caves mouth, and 219 bats in one hectare one kilometer from the cave. The bat count r km from the cave is the same in all directions. [Note that 1 km2 = 100 hectares, written 100 ha.] Give an overestimate of the number of bats between 3 and 4 km from the cave.

Problems 1012 refer to a colony of bats which flies out of a cave each night to eat insects. To estimate the colonys size, a naturalist counts samples of bats at different distances from the cave. Table 8.3 gives n, her count per hectare, at a distance r km from the cave. For instance, she counts 300 bats in one hectare at the caves mouth, and 219 bats in one hectare one kilometer from the cave. The bat count r km from the cave is the same in all directions. [Note that 1 km2 = 100 hectares, written 100 ha.] Give an underestimate of the number of bats between 3 and 4 km from the cave.

Problems 1012 refer to a colony of bats which flies out of a cave each night to eat insects. To estimate the colonys size, a naturalist counts samples of bats at different distances from the cave. Table 8.3 gives n, her count per hectare, at a distance r km from the cave. For instance, she counts 300 bats in one hectare at the caves mouth, and 219 bats in one hectare one kilometer from the cave. The bat count r km from the cave is the same in all directions. [Note that 1 km2 = 100 hectares, written 100 ha.] Letting n = f(r), write an integral in terms of f representing the number of bats in the cave. Assume that bats fly no farther away than 5 km from the cave. Do not evaluate the integral.

Find the total mass of the triangular region in Figure 8.64, which has density (x)=1+ x grams/cm2.

A rectangular plate is located with vertices at points (0, 0),(2, 0),(2, 3) and (0, 3) in the xy-plane. The density of the plate at point (x, y) is (y) = 2+y2 gm/cm2 and x and y are in cm. Find the total mass of the plate.

Circle City, a typical metropolis, is densely populated near its center, and its population gradually thins out toward the city limits. In fact, its population density is 10,000(3 r) people/square mile at distance r miles from the center. (a) Assuming that the population density at the city limits is zero, find the radius of the city. (b) What is the total population of the city?

The density of oil in a circular oil slick on the surface of the ocean at a distance r meters from the center of the slick is given by (r) = 50/(1 + r) kg/m2. (a) If the slick extends from r = 0 to r = 10,000 m, find a Riemann sum approximating the total mass of oil in the slick. (b) Find the exact value of the mass of oil in the slick by turning your sum into an integral and evaluating it. (c) Within what distance r is half the oil of the slick contained?

The soot produced by a garbage incinerator spreads out in a circular pattern. The depth, H(r), in millimeters, of the soot deposited each month at a distance r kilometers from the incinerator is given by H(r)=0.115e2r. (a) Write a definite integral giving the total volume of soot deposited within 5 kilometers of the incinerator each month. (b) Evaluate the integral you found in part (a), giving your answer in cubic meters.

A cardboard figure has the shape shown in Figure 8.65. The region is bounded on the left by the line x = a, on the right by the line x = b, above by f(x), and below by g(x). If the density (x) gm/cm2 varies only with x, find an expression for the total mass of the figure, in terms of f(x), g(x), and (x).

The storage shed in Figure 8.66 is the shape of a halfcylinder of radius r and length l. (a) What is the volume of the shed? (b) The shed is filled with sawdust whose density (mass/unit volume) at any point is proportional to the distance of that point from the floor. The constant of proportionality is k. Calculate the total mass of sawdust in the shed.

The following table gives the density D (in gm/cm3) of the earth at a depth x km below the earths surface. The radius of the earth is about 6370 km. Find an upper and a lower bound for the earths mass such that the upper bound is less than twice the lower bound. Explain your reasoning; in particular, what assumptions have you made about the density?

Water leaks out of a tank through a square hole with 1- inch sides. At time t (in seconds) the velocity of water flowing through the hole is v = g(t) ft/sec. Write a defi- nite integral that represents the total amount of water lost in the first minute.

An exponential model for the density of the earths atmosphere says that if the temperature of the atmosphere were constant, then the density of the atmosphere as a function of height, h (in meters), above the surface of the earth would be given by (h)=1.28e0.000124h kg/m3. (a) Write (but do not evaluate) a sum that approximates the mass of the portion of the atmosphere from h = 0 to h = 100 m (i.e., the first 100 meters above sea level). Assume the radius of the earth is 6400 km. (b) Find the exact answer by turning your sum in part (a) into an integral. Evaluate the integral.

Three point masses of 4 gm each are placed at x = 6, 1 and 3. Where should a fourth point mass of 4 gm be placed to make the center of mass at the origin?

A rod of length 3 meters with density (x) = 1+ x2 grams/meter is positioned along the positive x-axis, with its left end at the origin. Find the total mass and the center of mass of the rod.

A rod with density (x) = 2 + sin x lies on the x-axis between x = 0 and x = . Find the center of mass of the rod.

A rod of length 1 meter has density (x) = 1+ kx2 grams/meter, where k is a positive constant. The rod is lying on the positive x-axis with one end at the origin. (a) Find the center of mass as a function of k. (b) Show that the center of mass of the rod satisfies 0.5 < x < 0.75.

A rod of length 2 meters and density (x)=3 ex kilograms per meter is placed on the x-axis with its ends at x = 1. (a) Will the center of mass of the rod be on the left or right of the origin? Explain. (b) Find the coordinate of the center of mass.

One half of a uniform circular disk of radius 1 meter lies in the xy-plane with its diameter along the y-axis, its center at the origin, and x > 0. The mass of the half-disk is 3 kg. Find (x, y).

A metal plate, with constant density 2 gm/cm2, has a shape bounded by the curve y = x2 and the x-axis, with 0 x 1 and x, y in cm. (a) Find the total mass of the plate. (b) Sketch the plate, and decide, on the basis of the shape, whether x is less than or greater than 1/2. (c) Find x.

A metal plate, with constant density 5 gm/cm2, has a shape bounded by the curve y = x and the x-axis, with 0 x 1 and x, y in cm. (a) Find the total mass of the plate. (b) Find x and y.

An isosceles triangle with uniform density, altitude a, and base b is placed in the xy-plane as in Figure 8.67. Show that the center of mass is at x = a/3, y = 0. Hence show that the center of mass is independent of the triangles base.

Find the center of mass of a cone of height 5 cm and base diameter 10 cm with constant density gm/cm3.

A solid is formed by rotating the region bounded by the curve y = ex and the x-axis between x = 0 and x = 1, around the x-axis. It was shown in Example 1 on page 422 that the volume of this solid is (1 e2)/2. Assuming the solid has constant density , find x and y.

(a) Find the mass of a pyramid of constant density gm/cm3 with a square base of side 40 cm and height 10 cm. [That is, the vertex is 10 cm above the center of the base.] (b) Find the center of mass of the pyramid.

In Problems 3538, explain what is wrong with the statement. A 10 cm rod can have mass density given by f(x) = x2 5x gm/cm, at a point x cm from one end.

In Problems 3538, explain what is wrong with the statement. The center of mass of a rod with density x2 gm/cm for 0 x 10 is given by - 10 0 x3 dx.

In Problems 3538, explain what is wrong with the statement. If the center of mass of a rod is in the center of the rod, then the density of the rod is constant.

In Problems 3538, explain what is wrong with the statement. A disk with radius 3 cm and density (r)=3 r gm/cm2, where r is in centimeters from the center of the disk, has total mass 27 gm.

In Problems 3941, give an example of: A mass density on a rod such that the rod is most dense at one end but the center of mass is nearer the other end.

In Problems 3941, give an example of: A rod of length 2 cm, whose density (x) makes the center of mass not at the center of the rod.

In Problems 3941, give an example of: A rod of length 2 cm, whose density (x) makes the center of mass at the center of the rod.

In Problems 4248, are the statements true or false? Give an explanation for your answer To find the total population in a circular city, we always slice it into concentric rings, no matter what the population density function.

In Problems 4248, are the statements true or false? Give an explanation for your answer A city occupies a region in the xy-plane, with population density (y)=1+ y. To set up an integral representing the total population in the city, we should slice the region parallel to the y-axis.

In Problems 4248, are the statements true or false? Give an explanation for your answer The population density in a circular city of radius 2 depends on the distance r from the center by f(r) = 10 3r, so that the density is greatest at the center. Then the population of the inner city, 0 r 1, is greater than the population of the suburbs, 1 r 2.

In Problems 4248, are the statements true or false? Give an explanation for your answer The location of the center of mass of a system of three masses on the x-axis does not change if all the three masses are doubled.

In Problems 4248, are the statements true or false? Give an explanation for your answer The center of mass of a region in the plane cannot be outside the region.

In Problems 4248, are the statements true or false? Give an explanation for your answer Particles are shot at a circular target. The density of particles hitting the target decreases with the distance from the center. To set up a definite integral to calculate the total number of particles hitting the target, we should slice the region into concentric rings.

In Problems 4248, are the statements true or false? Give an explanation for your answer A metal rod of density f(x) lying along the x-axis from x = 0 to x = 4 has its center of mass at x = 2. Then the two halves of the rod on either side of x = 2 have equal mass.