Evaluate ∫ from 0 to 1 of 1/(1+x^2) dx by using Simpson's 3/8 rule.

Steps to Solve

1. Identify the integral and function We need to evaluate the integral

$$ \int_0^1 \frac{1}{1+x^2} , dx $$

using Simpson's 3/8 rule.

2. Determine the number of intervals (n) For Simpson's 3/8 rule, we need ( n ) to be a multiple of 3. In this case, let’s use ( n = 3 ) over the interval [0, 1]. This gives us ( h = \frac{b - a}{n} = \frac{1 - 0}{3} = \frac{1}{3} ).

3. Calculate the points The points at which we will evaluate the function are

• ( x_0 = 0 )
• ( x_1 = h = \frac{1}{3} )
• ( x_2 = 2h = \frac{2}{3} )
• ( x_3 = 1 )

4. Evaluate the function at these points Calculate ( f(x) = \frac{1}{1 + x^2} ) at these points:

• ( f(0) = \frac{1}{1+0^2} = 1 )
• ( f\left(\frac{1}{3}\right) = \frac{1}{1+\left(\frac{1}{3}\right)^2} = \frac{1}{1+\frac{1}{9}} = \frac{1}{\frac{10}{9}} = \frac{9}{10} )
• ( f\left(\frac{2}{3}\right) = \frac{1}{1+\left(\frac{2}{3}\right)^2} = \frac{1}{1+\frac{4}{9}} = \frac{1}{\frac{13}{9}} = \frac{9}{13} )
• ( f(1) = \frac{1}{1+1^2} = \frac{1}{2} )

5. Apply Simpson's 3/8 rule formula The formula for Simpson's 3/8 rule is:

$$ I \approx \frac{3h}{8} \left[ f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3) \right] $$

Substituting in our values:

$$ I \approx \frac{3 \cdot \frac{1}{3}}{8} \left[ 1 + 3 \cdot \frac{9}{10} + 3 \cdot \frac{9}{13} + \frac{1}{2} \right] $$

6. Calculate the integral value First calculate the coefficients:

$$ I \approx \frac{1}{8} \left[ 1 + \frac{27}{10} + \frac{27}{13} + \frac{1}{2} \right] $$

Finding a common denominator (130) for simplification:

• ( 1 = \frac{130}{130} )
• ( \frac{27}{10} = \frac{351}{130} )
• ( \frac{27}{13} = \frac{270}{130} )
• ( \frac{1}{2} = \frac{65}{130} )

So:

$$ I \approx \frac{1}{8} \left[ \frac{130 + 351 + 270 + 65}{130} \right] = \frac{1}{8} \left[ \frac{816}{130} \right] $$

Now simplifying:

$$ I \approx \frac{816}{1040} = \frac{204}{260} = \frac{102}{130} = \frac{51}{65} $$

7. Final value The estimated value of the integral is

$$ \frac{51}{65} $$