Multi-Variable Calculus Assignment

due Tuesday, April 3, 2001

Exercises

1. Evaluate the double integral

∫ ∫_{D} x^{2}y^{2} dxdy ,

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 (x̄, ȳ), where

x̄ = \frac{1}{A(S)} ∫ ∫_{S} x dxdy
ȳ = \frac{1}{A(S)} ∫ ∫_{S} y dxdy

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 = −3x.