Example 2

A circular disc of radius r has density at each point equal to the distance of the point from the y-axis. Find its mass. (The center of the circle, shown in Figure 6.6.4, is at the origin.) The circle is the region between the curves - and from -r to r. The density at a point (x, y) in the disc is |x|. By symmetry, all four quadrants have the same mass. We shall find the mass m₁ of the first quadrant and multiply by four.

Put

u = r² - x², du = -2xdx; u = r²

when x = 0, and u = 0 when x = r.

Then m = 4m₁ = (4/3)r³.

Figure 6.6.4