Math 21D - Practice Midterm 1
Problem 1
Sketch the region of integration, reverse the order of integration, and evaluate
the integral.
Z1
0Z1
√y
ex3dxdy
Problem 2
Calculate the center of mass of the region in the first quadrant bounded by the
circles r= 3 and r= 2. Assume that the density is constant: δ= 1.
Problem 3
Calculate the volume of the solid bounded by z= 0, z=px2+y2, and the
cylinder (x−1)2+y2= 1.
Problem 4
Calculate the volume of the solid bounded below by z= 0 and above by the
surfaces x2+y2+z2= 4 and z=px2+y2.
Problem 5
Evaluate the double integral
Z2
0Z2−x
0
(y−x)2(x+y)4dydx
by using the transformation
x=u−v
2y=u+v
2
Problem 6
Find the radius of gyration about the z axis Izof the cylinder of constant density
δ= 1 bounded by x2+y2= 4, z= 0, and z=π.