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
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)
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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 ...
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.