GATE (TF) Textile 2017 Question Paper Solution

Question 27 (Textile Engineering & Fibre Science)

Using Simpson’s 1/3 rule, the value of the integral $\frac{1}{\pi}\int_{0}^{\pi}(1+\sqrt{sin(x)}\;)dx$ . accurate to two decimal places is __1.65 to 1.76__.

Answer / Solution

Given , f(x)= $\frac{1}{\pi}\int_{0}^{\pi}(1+\sqrt{sin(x)}\;)dx$
Let devide the whole interval in to 4 equal parts.
i.e., h= $\frac{\pi-0}{4}$
h= $\frac{\pi}{4}$

x	0	$\frac{\pi}{4}$	$\frac{\pi}{2}$	$\frac{3\pi}{4}$	$\pi$
y	$\frac{1}{\pi}$	$\frac{1.84}{\pi}$	$\frac{2}{\pi}$	$\frac{1.84}{\pi}$	$\frac{1}{\pi}$
	yo	y1	y2	y3	y4

By 1/3 rd Simpson’s rule-
= $\frac{h}{3}[(y_o+y_n)+2(y_e)+4(y_o)]$
= $\frac{\pi}{3\times4}[(y_o+y_4)+2(y_2)+4(y_1+y_3)]$
= $\frac{\pi}{3\times4}[(\frac{1}{\pi}+\frac{1}{\pi})+2(\frac{2}{\pi})+4(\frac{1.84}{\pi}+\frac{1.84}{\pi})]$
=1.73 (Ans)