Let D be the annular region 2 2 , a r b where b>a> 0. Assume that D has a uniform charge

What is the mass density (x, y, z) of a solid of volume 5 m3 with uniform mass density and total mass 25 kg?

A domain D in R2 with uniform mass density is symmetric with respect to the y-axis. Which of the following are true? (a) xCM = 0 (b) yCM = 0 (c) Ix = 0 (d) Iy = 0

If p(x, y) is the joint probability density function of random variables X and Y , what does the double integral of p(x, y) over [0, 1][0, 1] represent? What does the integral of p(x, y) over the triangle bounded by x = 0, y = 0, and x + y = 1 represent?

Find the total population within a 4-km radius of the city center (located at the origin) assuming a population density of (x, y) = 2000(x2 + y2)0.2 people per square kilometer.

Find the total population within the sector 2|x| y 8 assuming a population density of (x, y) = 100e0.1y people per square kilometer.

Find the total mass of the solid region W defined by x 0, y 0, x2 + y2 4, and x z 32 x (in centimeters) assuming a mass density of (x, y, z) = 6y g/cm3.

Calculate the total charge of the solid ball x2 + y2 + z2 5 (in centimeters) assuming a charge density (in coulombs per cubic centimeter) of (x, y, z) = (3 108)(x2 + y2 + z2) 1/2

Compute the total mass of the plate in Figure 10 assuming a mass density of f (x, y) = x2/(x2 + y2) g/cm2.

Assume that the density of the atmosphere as a function of altitude h (in kilometers) above sea level is (h) = aebh kg/km3, where a = 1.225 109 and b = 0.13. Calculate the total mass of the atmosphere contained in the cone-shaped region  x2 + y2 h 3.

Calculate the total charge on a plate D in the shape of the ellipse with the polar equation r2 = 1 6 sin2 + 1 9 cos2 1 with the disk x2 + y2 1 removed (Figure 11) assuming a charge density of (r, ) = 3r4 C/cm2.

In Exercises 1114, find the centroid of the given region assuming the density (x, y) = 1. Region bounded by y = 1 x2 and y = 0

In Exercises 1114, find the centroid of the given region assuming the density (x, y) = 1. Region bounded by y2 = x + 4 and x = 4

In Exercises 1114, find the centroid of the given region assuming the density (x, y) = 1. Quarter circle x2 + y2 R2, x 0, y 0

In Exercises 1114, find the centroid of the given region assuming the density (x, y) = 1. Infinite lamina bounded by the x- and y-axes and the graph of y = ex

Use a computer algebra system to compute numerically the centroid of the shaded region in Figure 12 bounded by r2 = cos 2

Show that the centroid of the sector in Figure 13 has y-coordinate y = 2R 3 sin

In Exercises 1719, find the centroid of the given solid region assuming a density of (x, y) = 1. Hemisphere x2 + y2 + z2 R2, z 0

In Exercises 1719, find the centroid of the given solid region assuming a density of (x, y) = 1. Region bounded by the xy-plane, the cylinder x2 + y2 = R2, and the plane x/R + z/H = 1, where R > 0 and H > 0

In Exercises 1719, find the centroid of the given solid region assuming a density of (x, y) = 1. The ice cream cone region W bounded, in spherical coordinates, by the cone = /3 and the sphere = 2

Show that the z-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane x a + y b + z c = 1 in Figure 14 is z = c/4. Conclude by symmetry that the centroid is (a/4, b/4, c/4).

Find the centroid of the region W lying above the unit sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 x2 y2 (Figure 15)

Let R > 0 and H > 0, and let W be the upper half of the ellipsoid x2 + y2 + (Rz/H )2 = R2, where z 0 (Figure 16). Find the centroid of W and show that it depends on the height H but not on the radius R.

In Exercises 2326, find the center of mass of the region with the given mass density . Region bounded by y = 4 x, x = 0, y = 0; (x, y) = x

In Exercises 2326, find the center of mass of the region with the given mass density . Region bounded by y2 = x + 4 and x = 0; (x, y) = |y|

In Exercises 2326, find the center of mass of the region with the given mass density . Region |x|+|y| 1; (x, y) = (x + 1)(y + 1)

In Exercises 2326, find the center of mass of the region with the given mass density . Semicircle x2 + y2 R2, y 0; (x, y) = y

Find the z-coordinate of the center of mass of the first octant of the unit sphere with mass density (x, y, z) = y (Figure 17).

Find the center of mass of a cylinder of radius 2 and height 4 and mass density e-z,where z is the eight above the base

Let R be the rectangle [a, a][b, b] with uniform density and total mass M. Calculate: (a) The mass density of R (b) Ix and I0 (c) The radius of gyration about the x-axis

Calculate Ix and I0 for the rectangle in Exercise 29 assuming a mass density of (x, y) = x.

Calculate I0 and Ix for the disk D defined by x2 + y2 16 (in meters), with total mass 1000 kg and uniform mass density. Hint: Calculate I0 first and observe that I0 = 2Ix . Express your answer in the correct units

Calculate Ix and Iy for the half-disk x2 + y2 R2, x 0 (in meters), with total mass M kilograms and uniform mass density.

In Exercises 3336, let D be the triangular domain bounded by the coordinate axes and the line y = 3 x, with mass density (x, y) = y. Compute the given quantities. 33. Total mass

In Exercises 3336, let D be the triangular domain bounded by the coordinate axes and the line y = 3 x, with mass density (x, y) = y. Compute the given quantities.Center of mass

In Exercises 3336, let D be the triangular domain bounded by the coordinate axes and the line y = 3 x, with mass density (x, y) = y. Compute the given quantities.Ix

In Exercises 3336, let D be the triangular domain bounded by the coordinate axes and the line y = 3 x, with mass density (x, y) = y. Compute the given quantities.I0

In Exercises 3740, let D be the domain between the line y = bx/a and the parabola y = bx2/a2, where a, b > 0.Assume the mass density is (x, y) = 1 for Exercise 37 and (x, y) = xy for Exercises 3840. Compute the given quantities. 37. Centroid

In Exercises 3740, let D be the domain between the line y = bx/a and the parabola y = bx2/a2, where a, b > 0.Assume the mass density is (x, y) = 1 for Exercise 37 and (x, y) = xy for Exercises 3840. Compute the given quantities.Center of mass

In Exercises 3740, let D be the domain between the line y = bx/a and the parabola y = bx2/a2, where a, b > 0.Assume the mass density is (x, y) = 1 for Exercise 37 and (x, y) = xy for Exercises 3840. Compute the given quantities.Ix

In Exercises 3740, let D be the domain between the line y = bx/a and the parabola y = bx2/a2, where a, b > 0.Assume the mass density is (x, y) = 1 for Exercise 37 and (x, y) = xy for Exercises 3840. Compute the given quantities.I0

Calculate the moment of inertia Ix of the disk D defined by x2 + y2 R2 (in meters), with total mass M kg. How much kinetic energy (in joules) is required to rotate the disk about the x-axis with angular velocity 10 rad/s?

Calculate the moment of inertia Iz of the box W = [a, a] [a, a][0, H] assuming that W has total mass M.

Show that the moment of inertia of a sphere of radius R of total mass M with uniform mass density about any axis passing through the center of the sphere is 2 5MR2. Note that the mass density of the sphere is = M/ 4 3R3 .

Use the result of Exercise 43 to calculate the radius of gyration of a uniform sphere of radius R about any axis through the center of the sphere

In Exercises 45 and 46, prove the formula for the right circular cylinder in Figure 18. 45. Iz = 1 2MR2

In Exercises 45 and 46, prove the formula for the right circular cylinder in Figure 18.Ix = 1 4MR2 + 1 12MH2

The yo-yo in Figure 19 is made up of two disks of radius r = 3 cm and an axle of radius b = 1 cm. Each disk has mass M1 = 20 g, and the axle has mass M2 = 5 g.(a) Use the result of Exercise 45 to calculate the moment of inertia I of the yo-yo with respect to the axis of symmetry. Note that I is the sum of the moments of the three components of the yo-yo. (b) The yo-yo is released and falls to the end of a 100-cm string, where it spins with angular velocity . The total mass of the yo-yo is m = 45 g, so the potential energy lost is PE = mgh = (45)(980)100 g-cm2/s2. Find using the fact that the potential energy is the sum of the rotational kinetic energy and the translational kinetic energy and that the velocity v = b since the string unravels at this rate.

Calculate Iz for the solid region W inside the hyperboloid x2 + y2 = z2 + 1 between z = 0 and z = 1.

Calculate P (0 X 2; 1 Y 2), where X and Y have joint probability density function p(x, y) =  1 72 (2xy + 2x + y) if 0 x 4 and 0 y 2 0 otherwise

Calculate the probability that X + Y 2 for random variables with joint probability density function as in Exercise 49.

The lifetime (in months) of two components in a certain device are random variables X and Y that have joint probability distribution function p(x, y) =  1 9216 (48 2x y) if x 0, y 0, 2x + y 48 0 otherwise Calculate the probability that both components function for at least 12 months without failing. Note that p(x, y)is nonzero only within the triangle bounded by the coordinate axes and the line 2x + y = 48 shown in Figure 20. 2x + y = 48 x (months) 12 24 Region where x 12, y 12 and 2x + y 48 36 24 12 48 y (months)

Find a constant C such that p(x, y) = " Cxy if 0 x and 0 y 1 x 0 otherwise is a joint probability density function. Then calculate:

Find a constant C such that p(x, y) = " Cy if 0 x 1 and x2 y x 0 otherwise is a joint probability density function. Then calculate the probability that Y X3/2.

Numbers X and Y between 0 and 1 are chosen randomly. The joint probability density is p(x, y) = 1 if 0 x 1 and 0 y 1, and p(x, y) = 0 otherwise. Calculate the probability P that the product XY is at least 1 2 .

According to quantum mechanics, the x- and y-coordinates of a particle confined to the region R = [0, 1][0, 1] are random variables with joint probability density function p(x, y) = " C sin2(2x)sin2(2ny) if (x, y) R 0 otherwise The integers  and n determine the energy of the particle, and C is a constant. (a) Find the constant C. (b) Calculate the probability that a particle with  = 2, n = 3 lies in the region  0, 1 4   0, 1 8  .

The wave function for the 1s state of an electron in the hydrogen atom is 1s() = 1  a3 0 e/a0 where a0 is the Bohr radius. The probability of finding the electron in a region W of R3 is equal to W p(x, y, z) dV where, in spherical coordinates, p() = |1s()| 2 Use integration in spherical coordinates to show that the probability of finding the electron at a distance greater than the Bohr radius is equal to 5/e2 0.677. (The Bohr radius is a0 = 5.3 1011 m, but this value is not needed.) 5

According to Coulombs Law, the attractive force between two electric charges of magnitude q1 and q2 separated by a distance r is kq1q2/r2 (k is a constant). Let F be the net force on a charged particle P of charge Q coulombs located d centimeters above the center of a circular disk of radius R, with a uniform charge distribution of density coulombs per square meter (Figure 21). By symmetry, F acts in the vertical direction. (a) Let R be a small polar rectangle of size r located at distance r. Show that R exerts a force on P whose vertical component is kQd (r2 + d2)3/2 r r (b) Explain why F is equal to the following double integral, and evaluate: F = kQd 2 0 R 0 r dr d (r2 + d2)3/2 FIGUR

Let D be the annular region 2 2 , a r b where b>a> 0. Assume that D has a uniform charge distribution of coulombs per square meter. Let F be the net force on a charged particle of charge Q coulombs located at the origin (by symmetry, F acts along the x-axis). (a) Argue as in Exercise 57 to show that F = kQ /2 =/2 b r=a cos r2 r dr d (b) Compute F. F

Let D be the domain in Figure 22.Assume that D is symmetric with respect to the y-axis; that is, both g1(x) and g2(x) are even functions. (a) Prove that the centroid lies on the y-axisthat is, that x = 0. (b) Show that if the mass density satisfies (x, y) = (x, y), then My = 0 and xCM = 0. y = g2 (x) y = g1(x) x

Pappuss Theorem Let A be the area of the region D between two graphs y = g1(x) and y = g2(x) over the interval [a, b], where g2(x) g1(x) 0. Prove Pappuss Theorem: The volume of the solid obtained by revolving D about the x-axis is V = 2Ay, where y is the y-coordinate of the centroid of D (the average of the y-coordinate). Hint: Show that Ay = b x=a g2(x) y=g1(x) y dy dx 61

Use Pappuss Theorem in Exercise 60 to show that the torus obtained by revolving a circle of radius b centered at (0, a) about the x-axis (where b

Use Pappuss Theorem to compute y for the upper half of the disk x2 + y2 a2, y 0.

Parallel-Axis Theorem Let W be a region in R3 with center of mass at the origin. Let Iz be the moment of inertia of W about the zaxis, and let Ih be the moment of inertia about the vertical axis through a point P = (a, b, 0), where h =  a2 + b2. By definition, Ih = W ((x a)2 + (y b)2)(x, y, z) dV Prove the Parallel-Axis Theorem: Ih = Iz + Mh2.

Let W be a cylinder of radius 10 cm and height 20 cm, with total mass M = 500 g. Use the Parallel-Axis Theorem (Exercise 63) and the result of Exercise 45 to calculate the moment of inertia of W about an axis that is parallel to and at a distance of 30 cm from the cylinders axis of symmetry.