Consider a rectangular prism [0, 10]×[0, 2]×[0, 3] (all units in cm) with density function is...
Consider a rectangular prism [0, 10]×[0, 2]×[0, 3] (all units in cm) with density function is δ(x, y, z) = (0.5)y + (0.1)z + 0.2.
1. Set up the integral for the mass of the object.
2. Set up the integral for the moment with respect to the xy-plane (i.e. in the z direction).
Homework Answers
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Consider a rectangular prism [0, 10]×[0, 2]×[0, 3] (all units in
cm) with density function is...
11.1) a) Verify that the function f(x,y) given below is a joint density function for r...
11.1) a) Verify that the function f(x,y) given below is a joint density function for r and y: ſ4.ty if 0 <r<1, 0 <y<1 f(x, y) = { 10 otherwise b) For the probability density function above, find the probability that r is greater than 1/2 and y is less than 1/3. 11.2) For the same probability density function f(x,y) as from Problem #1. Find the expected values of r and y. 11.3) a) Let R= [0,5] x [0,2]. For...
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and...
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and height 6 and density function f(x,y,z) d density function f(x, y, z) (set-up triple integral then evaluate by hand) x2 + y2 +1 2 ty +1
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and height 6 and density function f(x,y,z) d density function f(x, y, z) (set-up triple integral then...
5. Set up the iterated integral for evaluating SSS, f(r,0,z)dzrdrde over the given region D. D...
5. Set up the iterated integral for evaluating SSS, f(r,0,z)dzrdrde over the given region D. D is the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = 2 – y. z 2 z = 2 - y y = x
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 poi...
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
ketch the graph of the probability density function over the indicated interval. 2x 9 [0, 3]...
ketch the graph of the probability density function over the indicated interval. 2x 9 [0, 3] y y 0.7 0.7 0.6 0.6 0.51 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2 3 y у 0.7 0.7 0.6 0.6 0.5 0.54 0.4 0.41 0.3 0.3 0.2 0.2 y 0.71 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2 Find the indicated probabilities. (a) PO < x < 2) (b) P(1 < x < 2)...
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0...
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
A rectangular loop of wire with sides H = 23 cm and W = 78 cm is located in a region
A rectangular loop of wire with sides H = 23 cm and W = 78 cm is located in a region containing a constant magnetic field B = 1.34 T that is aligned with the positive y-axis as shown. The loop carries current I = 227 mA. The plane of the loop is inclined at an angle θ = 21° with respect to the x-axis. 1) What is μx, the x-component of the magnetic moment vector of the loop? 2) What is...
(2) Consider a thin plate with constant density 8 covering the region below the...
(2) Consider a thin plate with constant density 8 covering the region below the curve y = above the z-axis, and left of the line r = 9. r, Set up integrals that will give the mass of the plate, the moment about the z-axis, and the moment about the y-axis. Calculate the center of mass of the plate.
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based...
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based on the principle: rf (x) dr ES Where S is the set of values on which the function is defined i Compute the median y where: f(z) dz = Where m is the minimum value on which the function is defined.
Consider the following probability density function:...
A rectangular loop of wire with sides H = 28 cm and W = 52 cm...
A rectangular loop of wire with sides H = 28 cm and W = 52 cm is
located in a region containing a constant magnetic field B = 0.9 T
that is aligned with the positive y-axis as shown. The loop carries
current I = 177 mA. The plane of the loop is inclined at an angle θ
= 35o with respect to the x-axis.
1) What is μx, the x-component of the magnetic moment
vector of the loop? A-m2...
11.1) a) Verify that the function f(x,y) given below is a joint density function for r and y: ſ4.ty if 0 <r<1, 0 <y<1 f(x, y) = { 10 otherwise b) For the probability density function above, find the probability that r is greater than 1/2 and y is less than 1/3. 11.2) For the same probability density function f(x,y) as from Problem #1. Find the expected values of r and y. 11.3) a) Let R= [0,5] x [0,2]. For...
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and height 6 and density function f(x,y,z) d density function f(x, y, z) (set-up triple integral then evaluate by hand) x2 + y2 +1 2 ty +1
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and height 6 and density function f(x,y,z) d density function f(x, y, z) (set-up triple integral then...
5. Set up the iterated integral for evaluating SSS, f(r,0,z)dzrdrde over the given region D. D is the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = 2 – y. z 2 z = 2 - y y = x
Suppose E is the half-cylinder described by 2y2 between z and the ry-plane where y 2 0. Suppose further that the density at each point in E is proportional to the distance from the z-axis. (a) (6 points) Find an expression for the mass of E as a triple integral. Then briefly explain why this integral is difficult to compute Suppose E is the half-cylinder described by r' +ґ-1 between z = 4 and the xy-plane where y > 0....
ketch the graph of the probability density function over the indicated interval. 2x 9 [0, 3] y y 0.7 0.7 0.6 0.6 0.51 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2 3 y у 0.7 0.7 0.6 0.6 0.5 0.54 0.4 0.41 0.3 0.3 0.2 0.2 y 0.71 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2 Find the indicated probabilities. (a) PO < x < 2) (b) P(1 < x < 2)...
4. Let X and Y have joint probability density function f(x,y) = 139264 oray3 if 0 < x, y < 4 and y> 4-1, otherwise. (a) Set up but do not compute an integral to find E(XY). (b) Let fx() be the marginal probability density function of X. Set up but do not compute an integral to find fx(x) when I <r54. (c) Set up but do not compute an integral to find P(Y > X).
(2) Consider a thin plate with constant density 8 covering the region below the curve y = above the z-axis, and left of the line r = 9. r, Set up integrals that will give the mass of the plate, the moment about the z-axis, and the moment about the y-axis. Calculate the center of mass of the plate.
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based on the principle: rf (x) dr ES Where S is the set of values on which the function is defined i Compute the median y where: f(z) dz = Where m is the minimum value on which the function is defined.
Consider the following probability density function:...
A rectangular loop of wire with sides H = 28 cm and W = 52 cm is
located in a region containing a constant magnetic field B = 0.9 T
that is aligned with the positive y-axis as shown. The loop carries
current I = 177 mA. The plane of the loop is inclined at an angle θ
= 35o with respect to the x-axis.
1) What is μx, the x-component of the magnetic moment
vector of the loop? A-m2...
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