# Multiple Intergration, Area, Center of Mass, Centroid and Jacobian

1.Given the region R bounded by y=2x+2 , 2y=x and 4.

a) Set up a double integral for finding the area of R.

b) Set up a double integral to find the volume of the solid above R but below the surface

f(x,y) 2+4x.

c) Setup a triple integral to find the volume of the solid above R but below the surface f(x,y)=-x^2 +4x.

d) Set up the integral to find the moment of the solid in part b) about the xy-plane.

e) Set up the integral for finding the surface area of f(x, y) = ?x^2 + 4x above R.

1) Find the mass of the solid from part c) if d(x,y,z) = 2x.

g) If we assume a lamina with the shape of R is of homogeneous density, flnd the centroid.

2. Given the solid bounded by the two spheres x2 + y2 + z2 =1 and x2 + y2 + z2 =9 and the upper nappe of the cone = 3(x2 + y2),

a) Set up the integral for finding the volume using cylindrical coordinates.

b) Set up the integral for finding the volume using spherical coordinates.

3. Given the integral below, use u and v substitution to change the variables. Assume that the the integral is to be evaluated over the region R bounded by x = 2y, y = 2x, x + y =1, and x + y =2. (Do not evaluate the integral)

Please see the attached file for the fully formatted problems.

Â© BrainMass Inc. brainmass.com March 4, 2021, 6:12 pm ad1c9bdddfhttps://brainmass.com/math/integrals/multiple-integration-area-center-mass-centroid-jacobian-34893

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

Solution to question 1.

a) The area S of R is

b) The volume V is given by

c) Denote the region by . Then we have

d) The moment is given by

e) The surface area S* is given by

f) The mass of solid from part c) is

g) Since there are many homogenous ...

#### Solution Summary

Probems involving Multiple Intergration, Area, Center of Mass, Centroid, Moment, Surface Area and Jacobian are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.