We practice setting up calculations for centers of mass and centroids.
Note: It is important to get as much of an intuitive sense as you can about what the double integrals (which appear around the 3:30 mark) mean, but do not worry about precisely what they represent.
- The term “centroid” refers to the geometric center of a region. Practically speaking, this means we may assume constant density (e.g., density ).
- First we compute the “mass” of the region, which in this case is simply the area between curves:
- Next we compute the moments about the and axes. This always involves multiplying the integrand above by and , respectively (note the reversal), where are the coordinates of the geometric center of a typical slice.
- Using as the slicing variable, slices are horizontalvertical and consequently the -coordinate of the geometric center of a slice is just (but note that this would be different if were the slicing variable). Thus
- The -coordinate of the geometric center of a slice will be the average of -coordinates at the top and bottom of a slice. Therefore Thus
- In this example, we should use as the slicing variable, so the roles of and are largely switched in comparison to the previous example.
- First compute the mass:
- In this case, since slices are horizontalvertical , so
- Likewise, is the average of -coordinates of endpoints of a slice. Thus Therefore
- To conclude,