8.3 Applications of Integral Calculus
1. (a) Find the centroid of the region bounded by the line and the parabola (we assume that the density of the enclosed region is 1).
The area of the enclosed region is
Due to the formula , we have
So the centroid is .
(b) Use Pappus’s theorem to find the volume of the solid obtained by rotating about -axis the region enclosed by the line and the parabola .
Using the result (a), we know that -coordinate of the centroid for the enclosed region is . Therefore, with the aid of Pappus's centroid theorem, the volume of rotating the region is .
2. Find the center of mass of the region bounded by the cycloid below and axis (we assume that the density of the enclosed region is 1).
By the symmetry principle, the center of mass must lie on the axis, so . The area of the region is computed as follows:
Therefore, recalling the formula for -coordinate of the center of mass is given as follows:
Summing up, the center of mass is
3. By using graphical tools, find the approximate center of mass of the region which is bounded by the , .
polar_plot(sqrt(2+cos(2*theta)), (0, 2*pi), fill=True).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2)
The center of mass of the region approximately .