Mathematics 41, section 1 – Multivariable Calculus Discussion 8 – Polar Double Integrals, Centers of Mass November 18, 1999

Background

Here’s a joke to try out on your friends:

Question: What is a polar bear? Answer: It’s a rectangular bear after the change of coordinates

x = r cos θ, y = r sin θ.

Yesterday, we saw that if f(x, y) was any function, and R was a region in the plane given in polar coordinates by

R = {(r, θ) : a ≤ r ≤ b, α ≤ θ ≤ β},

then the double integral

∫ ∫_{R} f dA = ∫_{α}^{β} ∫_{a}^{b} f(r cos θ, r sin θ) rdrdθ.

Today we will look at how polar coordinates can simplify the computations of things like centers of mass when the geometry of the region R is well adapted to that coordinate system. First, recall the average value of f on R is given by:

\frac{∫ ∫_{R} f(x, y) dA}{area(R)} = \frac{∫ ∫_{R} f(x, y) dA}{∫ ∫_{R} 1 dA}

In particular, if f = x, or f = y we would get

\begin{aligned}\overline{x} & = \frac{∫ ∫_{R} x dA}{∫ ∫_{R} 1 dA} \\ \overline{y} & = \frac{∫ ∫_{R} y dA}{∫ ∫_{R} 1 dA} \end{aligned}

and the point (\overline{x}, \overline{y}) can be thought of as the center of mass of the region R (it would be the center of mass of a thin plate in the shape of R if the material of the plate had constant density).

Discussion Questions

A) Compute the center of mass of the region R in the first and second quadrants bounded by the positive x-axis, the line y = −x, and an arc of the circle of radius 2 centered at the origin. Suggestion: think polar! Answer:

(\overline{x}, \overline{y}) = (\frac{8\sqrt{2}}{9π}, \frac{8\sqrt{2} + 16}{9π}) ≐ (.4001, .9660).

B) Suppose you had a thin plate in the shape of the region R from question A, but now the density of the material at each point was variable, given by the function ρ(x, y) = x^{2} + y^{2}. What double integrals would compute total mass of the plate and its center of mass? (Think of \overline{x} and \overline{y} as “weighted averages,” weighted by the density function.)

C) Next, suppose we wanted to determine the center of mass of an object that had thickness. For simplicity, suppose our object is a rectangular solid S = [a, b] × [c, d] × [e, f] in R^{3}, but suppose that the object has varying density ρ(x, y, z) at each point (x, y, z) ∈ S. How could you approximate the total mass and the coordinates of the center of mass?

Assignment

Group write-ups due Monday, November 22.