Centroid Math Notes

Question

Was there a typo in the notes? I noticed that when calculating the center of mass for y they do not square twice as shown in the formula. Can anyone confirm this?

Answer 1

The formula should be $\bar {y} = \frac 1A\int_a^b \frac 12 (f^2(x) - g^2(x)) \ dx$

Note that there is a similar structure to volume of revolution problems.

By washers $V = 2\pi \int_a^b \frac 12 (f^2(x) - g^2(x)) \ dx$

By shells $V = 2\pi \int_a^b x(f(x) - g(x)) \ dx$

Or you could even say $V = 2\pi A \bar y$ when revolving about the x axis and $V = 2\pi A \bar x$ when revolving about the y axis.

If you have learned double integration.

$\bar {y} = \frac 1A\int_a^b \int_{g(x)}^{f(x)} y \ dy\ dx\\ \bar {x} = \frac 1A \int_a^b \int_{g(x)}^{f(x)} x \ dy\ dx\\$