Multivariable Calculus (Math 214) Assignment

due November 10, 1999

1. Evaluate the double integral

   INT[INT[_{D} {x y} 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]] and \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.

3. Use the version of the chain rule presented in this course and the ordinary (first year calculus) version of the fundamental theorem of calculus to obtain a formula for the integral of a vector field F over a parameterized curve C given by r(t), a <= t <= b, in the special case where the vector field is the gradient of a scalar-valued function f, i.e., F = \nabla f.

4. It has been pointed out that the chain rule in a multi-variable context is more a theoretical tool than a computational tool. List the useful facts that we have found so far which follow from it.