Single Variable Calculus, Chapter 5, 5.2, Section 5.2, Problem 10

Estimate $\displaystyle \int^{\frac{\pi}{2}}_0 \cos ^4x dx, n = 4$ using Midpoint Rule

The width of each sub-intervals is given to be $\displaystyle \Delta x = \frac{\displaystyle \frac{\pi}{2} - 0}{4} = \frac{\pi}{8}$. So the endpoints of the four sub-intervals are $\displaystyle 0, \frac{\pi}{8}, \frac{\pi}{4}, \frac{3 \pi}{8}$ and $\displaystyle \frac{\pi}{2}$. Thus, the midpoints are $\displaystyle \left( \frac{\displaystyle 0 + \frac{\pi}{8} }{2}\right) = \frac{\pi}{16}, \left( \frac{\displaystyle \frac{\pi}{8} + \frac{\pi}{4}}{2} \right) = \frac{3 \pi}{16}, \left( \frac{\displaystyle \frac{\pi}{4} + \frac{3 \pi}{8}}{2} \right) = \frac{5 \pi}{16}, \left( \frac{3 \pi}{8} + \frac{\pi}{2} \right) = \frac{7 \pi}{16}$.

Therefore, the Midpoint Rule gives..



$\begin{equation} \begin{aligned} \displaystyle \int^{\frac{\pi}{2}}_0 \cos^4 x dx \approx & \Delta x \left[ f\left( \frac{\pi}{16} \right) + f \left( \frac{3 \pi}{16} \right) + f \left( \frac{5 \pi}{16} \right) + f \left( \frac{7 \pi}{16} \right) \right] \\ \\ \approx & \frac{\pi}{8} [0.9253 + 0.4780 + 0.0953 + 0.0014] \\ \\ \approx & 0.5891 \end{aligned} \end{equation}$