Question

Use the composite trapezoidal rule to find approximations to 𝐼 = ∫ 𝐬𝐢𝐧𝒙 𝒅𝒙 𝝅

𝟎

with n = 1, 2, 4, and 8. Then perform Romberg extrapolation on the results.

Answer 1

$f(x)=xsinx$

for n = 1:

$I=\pi(\frac{f(0)+f(\pi)}{2})=0$

for n = 2:

$I=\frac{\pi}{2}(\frac{f(0)+f(\pi/2)}{2}+\frac{f(\pi/2)+f(\pi)}{2})=\frac{\pi^2}{4}=2.465$

for n = 4:

$I=\frac{\pi}{4}(\frac{f(0)+f(\pi/4)}{2}+\frac{f(\pi/4)+f(\pi/2)}{2}+\frac{f(\pi/2)+f(3\pi/4)}{2}+\frac{f(3\pi/4)+f(\pi)}{2})=$

$=\frac{\pi}{4}(\pi sin(\pi/4)/4+\pi /2+3\pi sin(3\pi/4)/4)=2.975$

for n = 8:

$I=\frac{\pi}{8}(f(\pi/8)+f(\pi/4)+f(3\pi/8)+f(\pi/2)+f(5\pi/8)+f(3\pi/4)+f(7\pi/8))=$

$=1.232(0.048+0.177+0.346+0.5+0.578+0.530+0.336)=3.098$

Romberg extrapolation:

$R_{j,k}=R_{j,k-1}+\frac{R_{j,k-1}-R_{j-1,k-1}}{4^{k-1}-1}$

for n = 1:

$R_{1,1}=\pi(\frac{f(0)+f(\pi)}{2})=0$

for n = 2:

$R_{2,1}=\frac{\pi}{2}(\frac{f(0)+f(\pi/2)}{2}+\frac{f(\pi/2)+f(\pi)}{2})=2.465$

$R_{2,2}=R_{2,1}+\frac{R_{2,1}-R_{1,1}}{3}=3.287$

for n = 4:

$R_{3,1}=\frac{\pi}{4}(\frac{f(0)+f(\pi/4)}{2}+\frac{f(\pi/4)+f(\pi/2)}{2}+\frac{f(\pi/2)+f(3\pi/4)}{2}+\frac{f(3\pi/4)+f(\pi)}{2})=2.975$

$R_{3,2}=R_{3,1}+\frac{R_{3,1}-R_{2,1}}{3}=3.145$

$R_{3,3}=R_{3,2}+\frac{R_{3,2}-R_{2,2}}{15}=3.136$

for n = 8:

$R_{4,1}=\frac{\pi}{8}(f(\pi/8)+f(\pi/4)+f(3\pi/8)+f(\pi/2)+f(5\pi/8)+f(3\pi/4)+$

$+f(7\pi/8))=3.098$

$R_{4,2}=R_{4,1}+\frac{R_{4,1}-R_{3,1}}{3}=3.139$

$R_{4,3}=R_{4,2}+\frac{R_{4,2}-R_{3,2}}{15}=3.139$

$I=R_{4,4}=R_{4,3}+\frac{R_{4,3}-R_{3,3}}{63}=3.139$