# Observe that we get the information to do the sketch

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Observe that we get the information to do the sketch from the limits of integration of the original iterated integral, namely that the region, R, consists of the pairs ( x , y ) which satisfy sin -1 ( y ) x π /2 when 0 y 1. The bounding curves may be read from these inequalities easily. Of course in reversing the integrals, a key piece of the puzzle is that x = sin -1 ( y ) if, and only if y = sin( x ) for the critical ordered pairs along the top boundary. Thus, 0 y sin( x ) when we have 0 x π /2. ______________________________________________________________________ 4. (10 pts.) Set up, but do not attempt to evaluate, a triple iterated integral in cartesian coordinates that would be used to find the volume of the solid G bounded below by the elliptic paraboloid z = 4 x 2 + y 2 and above by the cylindrical surface z = 4 - 3 y 2 , [Hint: The upper and lower surfaces are on a platter with a cherry on top. Find the projection onto the xy-plane of the curve obtained when the two surfaces intersect.] or will do. When the two surfaces intersect, we have 4 x 2 + y 2 = 4 - 3 y 2 . Consequently, the projection on the xy-plane of the curve of intersection is given by the set of ordered pairs satisfying the equation x 2 + y 2 = 1, an equation for the unit circle centered at the origin of the xy-plane.

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TEST3/MAC2313 Page 3 of 5 ______________________________________________________________________ 5. (10 pts.) Write down the triple iterated integral in cylindrical coordinates that would be used to compute the volume of the solid G whose top is the plane z = 9 and whose bottom is the paraboloid z = x 2 + y 2 . Do not attempt to evaluate the integral. The projection of the line of intersection of the plane z = 9 and the paraboloid z = x 2 + y 2 is the circle with radius 3 centered at the origin. I imagine you might be able to picture this in your bio-computer.
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