Question 1. Use cylindrical coordinates to nd the mass of the solid of density ez which lies in the closed region ![292_Determine the mass of the hemisphere.png](https://www.expertsmind.com/CMSImages/292_Determine%20the%20mass%20of%20the%20hemisphere.png)
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) =![2374_Determine the mass of the hemisphere1.png](https://www.expertsmind.com/CMSImages/2374_Determine%20the%20mass%20of%20the%20hemisphere1.png)
Determine the mass of the hemisphere. Note that sphere is not in the upper half space z > 0.
Hint: The identity cos(2θ) = cos2θ - sin2θ may be useful.
Question 3. Determine whether or not the vector eld F(x; y; z) = 14xyi + (7x2+ 18yz)j + 9y2k is conservative. If it is conservative, nd a function f(x; y; z) such that
![1240_Determine the mass of the hemisphere2.png](https://www.expertsmind.com/CMSImages/1240_Determine%20the%20mass%20of%20the%20hemisphere2.png)
Question 4. Evaluate the line integral
![402_Determine the mass of the hemisphere3.png](https://www.expertsmind.com/CMSImages/402_Determine%20the%20mass%20of%20the%20hemisphere3.png)
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 = x2 , 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; 6x3
+ 18xy2 ).
(Hint: Work done by a force eld F along a curve C is given by the line integral![1119_Determine the mass of the hemisphere5.png](https://www.expertsmind.com/CMSImages/1119_Determine%20the%20mass%20of%20the%20hemisphere5.png)