Multi-Variable Calculus Assignment

due Tuesday, April 3, 2001

Exercises

1. Evaluate the double integral

   INT[INT[_{D} {x^{2} y^{2}} dx dy ]] ,

   where D is the disk defined by the inequality x^{2} + y^{2} <= a^{2}, for a given positive constant a.

2. The centroid of a planar region S is defined to be the point (\bar{x}, \bar{y}), where

   \bar{x}	=	{1}/{A(S)} INT[INT[_{S} x dx dy]]
   \bar{y}	=	{1}/{A(S)} INT[INT[_{S} y dx dy]]

   and A(S) is the area of S. Find the centroid of the region that is bounded by the parabola y = 4 - x^{2} and the line y = -3 x.

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