Chapter 9, Problem 2P

The profile of a gear tooth shown in Fig.P9.2 is approximated by the trigonometric equation $y\left(x\right)=\frac{k}{2}\left(1-\cos \frac{2\pi x}{l}\right)$.

1. Estimate the area A using eight rectangles of equal width $\left(\Delta x=l/8\right)$
2. ,i.e.,

$A={\displaystyle \underset{i=1}{\overset{8}{\sum }}y\left(x_i\right)\Delta x}$. (b)Calculate the exact area by integration, i.e.,

$A={\displaystyle \underset{0}{\overset{l}{\int }}y\left(x\right)dx}$.