Multi-Variable Calculus Assignment

due Tuesday, April 17, 2001

Terminology

Integrals of functions over sets and centroids of sets are always defined relative to the notion of measure for the set and its subsets. For a curve the appropriate notion of measure is arc length, while for a surface the appropriate notion of measure is surface area.

The integral of a scalar-valued function f over a piece of curve given parametrically by

t ⟼ r(t)

is found by integrating

f(r(t)) ||r′(t)||

over the interval of the t-line corresponding to the arc in question. If f is the constant 1, then the integral is simply the length of the corresponding arc.

The integral of a scalar-valued function f over a piece of surface in R^{3} given parametrically by

(u, v) ⟼ s(u, v)

is found by integrating

f(s(u, v)) ||\frac{∂s}{∂u} × \frac{∂s}{∂v} ||

over the appropriate region in the plane of the parameterizing point (u, v). If f is the constant 1, then the integral is simply the area of the corresponding piece of surface.

Exercises

1. Find the centroid of the arc of the helix given by

x = 4 cos t
y = 4 sin t
z = 3t

for 0 ≤ t ≤ π/2.

2. For a given positive constant a find the surface area and the centroid of the portion of the spherical surface

x^{2} + y^{2} + z^{2} = a^{2}

lying in the first octant of space.

Hint: Use the spherical coordinates “longitude” and “co-latitude” as parameters.