Multivariable Calculus (Math 214) Assignment

due November 22, 1999

1. Discussion

Recall that a point-in-space-valued function r(u, v) parameterizes a piece of surface in space when (u, v) varies in a region R. Then the integral of a scalar-valued function f in space over that piece of surface is given by the formula

INT[INT[_{S} f ]] = INT[INT[_{R} f(r(u, v)) ||{\partial S}/{\partial u}\times {\partial S}/{\partial v}|| d ud v]] .

When f is the constant function 1 its integral over that piece of surface is the surface area of that piece of surface. For this reason one calls the scalar differential expression

d sigma = ||{\partial S}/{\partial u}\times {\partial S}/{\partial v}|| dudv

the element of surface area. This is analogous to the practice for a parameterized path R(t) of calling the scalar differential expression

d s = ||R'(t)|| d t

the element of arc length. Both notions take curvedness into account.

In both cases the average value of a scalar-valued function f on one of these geometric objects is defined to be the integral of f divided by the measure of the object, which is the integral of the constant function 1,, i.e., its arc length or surface area, as appropriate.

The centroid of a piece of curve or surface is the point whose each coordinate is determined as the average value of the corresponding coordinate on all the points of the object. This is fully in harmony with previous notions of centroid. (Note that the centroid need not be a point of the object.)

2. Exercises

  1. How does one locate the centroid of a half ball, i.e., a ``solid hemisphere''?

  2. Find the average value of the function

    f(x, y) = x y

    on the first quadrant portion of the circle (not the disk) of radius a > 0 with center at the origin.

  3. Find the centroid of the first quadrant portion of the circle (not the disk) of radius a with center at the origin.

  4. Find the centroid of the portion in the first octant of space of the sphere of radius a with center at the origin.

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