Question #264471

Evaluate integral of a = 0 and b =π\2 sin(t)dt by applying Simpsons rule with four equal intervals


1
Expert's answer
2021-11-12T15:48:21-0500
abf(t)dt\displaystyle\int_{a}^{b}f(t)dt

Δt3(f(t0)+4f(t1)+2f(t2)+4f(t3)\approx\dfrac{\Delta t}{3}\big(f(t_0)+4f(t_1)+2f(t_2)+4f(t_3)

+2f(t4)+...+4f(tn2)+2f(tn1)+f(tn))+2f(t_4)+...+4f(t_{n-2})+2f(t_{n-1})+f(t_n)\big)

Δt=ban\Delta t=\dfrac{b-a}{n}

We have that f(t)=sint,a=0,b=π/2,n=4.f(t)=\sin t, a=0, b=\pi/2, n=4.

Therefore


Δt=π/204=π/8\Delta t=\dfrac{\pi/2-0}{4}=\pi/8

Divide the interval [0,π/2][0, \pi/2]  into n=4n=4 subintervals of the length Δt=π/8Δt=π/8 with the following endpoints: a=0,π/8,π/4,3π/8,π/2=b.a=0, \pi/8, \pi/4, 3\pi/8, \pi/2=b.

 Evaluate the function at these endpoints


f(t0)=f(0)=sin(0)=0f(t_0)=f(0)=\sin(0)=0


4f(t1)=4f(π/8)=4sin(π/8)4f(t_1)=4f(\pi/8)=4\sin(\pi/8)

1.530733729460359≈1.530733729460359


2f(t2)=2f(π/4)=2sin(π/4)=22f(t_2)=2f(\pi/4)=2\sin(\pi/4)=\sqrt{2}


1.414213562373095\approx1.414213562373095


4f(t3)=4f(3π/8)=4sin(3π/8)4f(t_3)=4f(3\pi/8)=4\sin(3\pi/8)

3.695518130045147≈3.695518130045147

f(t4)=f(π/2)=sin(π/2)=1f(t_4)=f(\pi/2)=\sin(\pi/2)=1

Therefore


0π/2sintdt\displaystyle\int_{0}^{\pi/2}\sin t dt

(π/12)(0+1.530733729460359\approx(\pi/12)(0+1.530733729460359

+1.414213562373095+1.414213562373095

+3.695518130045147+1)+3.695518130045147+1)

1.000134584974194\approx1.000134584974194

0π/2sintdt1.000134584974194\displaystyle\int_{0}^{\pi/2}\sin t dt\approx1.000134584974194


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