Assignment:
Q1. 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.
f) Find the mass of the solid from part c) if δ(x,y,z) = 2x.
g) If we assume a lamina with the shape of R is of homogeneous density, flnd the centroid.
Q2. Given the solid bounded by the two spheres x2 + y2 + z2 =1 and x2 + y2 + z2 =9 and the upper nappe of the cone z2= 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.
Q3. 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)
Provide complete and step by step solution for the question and show calculations and use formulas.