Question

Use Simpson's Rule with n6 to approximate r using the given equation. (In Section 11.5, you will be able to evaluate the integral using inverse trigonometric functions. Round your answer to four decimal places.) 1/2 6 J = - La vengea

Answer 1

According to Simpson's rule,

AC f(x)dx (f(x0) +4f(21) +2f (x2)+4f (23+2f (24) +...+2f (In-2) + 4f (In-1) + f(n)) $s

Here, a=0 b=1/2,    , n=6

6 f(x) = /1-22

$\small \therefore \bigtriangleup x=\frac{\frac{1}{2}-0}{6}=\frac{1}{12}$

Dividing the interval [0,1/2] into six intervals of length 1/12 , we get

11 1115 51 d = 0, 126°43'12'2

$\small f(x_{o})=f(a)=f(0)=\frac{6}{\sqrt{1-0}}=6$

$\small 4f(x_{1})=4f(\frac{1}{12})=\frac{288\sqrt{143}}{143}\approx 24.0838$

$\small 2f(x_{2})=2f(\frac{1}{6})=\frac{72\sqrt{35}}{35}\approx 12.1702$

$\small 4f(x_{3})=4f(\frac{1}{4})=\frac{32\sqrt{15}}{5}\approx 24.7871$

$\small 2f(x_{4})=2f(\frac{1}{3})=9\sqrt{2}\approx 12.7279$

$\small 4f(x_{5})=4f(\frac{5}{12})=\frac{288\sqrt{119}}{119}\approx 26.4009$

$\small f(x_{6})=f(b)=f(\frac{1}{2})=4\sqrt{3}\approx 6.9282$

$\small \frac{\Delta x}{3}=\frac{1}{36}$

$\small \pi=\frac{1}{36}(6+24.0838+12.1702+...+6.9282)=\mathbf{3.1416}$