Multi-Variable Calculus Assignment

due Friday, April 6, 2001

Exercises

1. Find the centroid of the first quadrant portion of the disk

x^{2} + y^{2} ≤ a^{2}

when a is a given positive constant.

2. Find the centroid of a solid cone having radius a and altitude h for given positive constants a and h.

Hint: Such a solid cone may be realized, up to congruence, as the set of points (x, y, z) in R^{3} for which 0 ≤ z ≤ h and

\frac{r}{a} + \frac{z}{h} ≤ 1 ,

where r^{2} = x^{2} + y^{2}. In this realization the lateral surface of the cone is the surface of revolution obtained by rotating the line

\frac{x}{a} + \frac{z}{h} = 1

about the z-axis. A point on that surface is characterized by

\frac{r}{a} + \frac{z}{h} = 1 .