Riemann sum

[janeann]

Use a riemann sum with 4 rectangles and the midpoint rule to estimate the area under sin(x) on (0,pi). Hint: the double or half angles can be expressed as sin(2theta)=2sin(theta)cos(theta) and sin(theta)=sqrt(1-cos(2theta))/2.

So far i have the change of x being (-pi/4) and xj being -jpi/4. so then i have the integral from 0 to pi sin(x)= the summation n=4 i=1 (-pi/4)(pij/4), then i get pi^2/16 being the area. Im not sure when midpoint would come in.

 

[galactus]

Each subinterval has width \(\displaystyle \frac{{\pi}-0}{4}=\frac{\pi}{4}\)

Since the midpoint rule is being used, the first point is at \(\displaystyle \frac{\pi}{8}\). Half way between 0 and Pi/4.

So, the first one is \(\displaystyle sin(\frac{\pi}{8})\), then pi/4 added thereafter.

second: \(\displaystyle sin(\frac{\pi}{8}+\frac{\pi}{4})\)

third: \(\displaystyle sin(\frac{\pi}{8}+\frac{\pi}{2})\)

fourth: \(\displaystyle sin(\frac{\pi}{8}+\frac{3\pi}{4})\)

So, we have \(\displaystyle \frac{\pi}{4}\left(sin(\frac{\pi}{8})+sin(\frac{\pi}{8}+\frac{\pi}{4})+sin(\frac{\pi}{8}+\frac{\pi}{2})+sin(\frac{\pi}{8}+\frac{3\pi}{4})\right)\)