Question 1. Use cylindrical coordinates to nd the mass of the solid of density e^{z} which lies in the closed region
Question 2. The density of a hemisphere of radius a (y 0) with its base centred at the origin is given by (x; y; z) =
Determine the mass of the hemisphere. Note that sphere is not in the upper half space z > 0.
Hint: The identity cos(2θ) = cos^{2}θ - sin^{2}θ may be useful.
Question 3. Determine whether or not the vector eld F(x; y; z) = 14xyi + (7x^{2}+ 18yz)j + 9y^{2}k is conservative. If it is conservative, nd a function f(x; y; z) such that
Question 4. Evaluate the line integral
where C is the curve from (0; 0) to (0; 2) consisting of the path from (0; 0) to (1; 1) along the parabola y = x^{2} , followed by the path along straight line from (1; 1) to (0; 2).
Question 5. A particle starts at the point (-4,0), moves along the x-axis to (0,0) and the y-axis to (0,4) and then along the arc y = to the starting point. Use Green's theorem to nd the work done on this particle by the force eld F(x; y) = (18x; 6x^{3}
+ 18xy^{2} ).
(Hint: Work done by a force eld F along a curve C is given by the line integral