Quiz (Double Integrals In Polar Coordinates)

Show all work and justify each step.

(1) Use polar coordinates to evaluate the iterated integrals.

    (a)

    (b)

    (c)
    

(2) Find the volume of the given solid region.

    (a) The solid region common to the cylinder and the ellipsoid
    
    (b) The solid region bounded above by the cone below by the plane and on both sides by the cylinder
    
    (c) The ice cream cone bounded by the hemisphere and the cone
    
(3) For a constant where the plane cuts off a "cap" from the hemisphere Use a double integral in polar coordinates to find the volume of the cap. For what values of is the volume of the cap half the volume of the hemisphere?

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