Determine the mass of the hemisphere, Mathematics

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

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

 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

 

Question 4. Evaluate the line integral

402_Determine the mass of the hemisphere3.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 =955_Determine the mass of the hemisphere4.png 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 integral1119_Determine the mass of the hemisphere5.png

Posted Date: 2/16/2013 4:57:37 AM | Location : United States







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