How do you Use Simpson's rule with #n=8# to approximate the integral #int_0^2root4(1+x^2)dx#?

Question

Charles Arocha · Accepted Answer

The answer is 2.41223163.

For any numerical approximation of a function, you always start with a table of values. For your problem, we have:

#a=0#
#b=2#
#n=8#

So,

#Delta x=(b-a)/n=1/4#
#x_i=a+i Delta x, i in {0, 1, ..., 8}#

Now it is a matter of applying Simpson's Rule:

#int_0^2 (1+x^2)^(1/4)dx = int_0^2 f(x)dx ~~ (Delta x)/3(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+2f(x_6)+4f(x_7)+f(x_8))#

I'll skip the substitution of values because it's messy.
We get 2.41223163 as the approximation.

Using numerical integration on a calculator gets a value of 2.412231919 which means the approximation is good to 6 decimal places.

Notice the pattern of the coefficients for the sum is: 1, 4, 2, 4, ..., 2, 4, 1. This means that to use Simpson's Rule, we need an odd number of values or an even number of intervals; #n# is even.

Aurora Alvarado · Answer

undefined To use Simpson's rule with $ n = 8 $ to approximate the integral $ \int_0^2 \sqrt{4(1+x^2)} \, dx $, follow these steps:

1. Divide the interval $[0, 2]$ into $ n = 8 $ subintervals of equal width. The width of each subinterval, $ \Delta x $, is calculated as $ \frac{b - a}{n} $, where $ a = 0 $ and $ b = 2 $.

2. Calculate the function values $ f(x_i) $ at each of the $ n+1 $ equally spaced points within the interval $[0, 2]$.

3. Apply Simpson's rule formula:
$$ \text{Approximate} = \frac{\Delta x}{3} \left[ f(x_0) + 4\sum_{i=1}^{n/2} f(x_{2i-1}) + 2\sum_{i=1}^{n/2-1} f(x_{2i}) + f(x_n) \right] $$

4. Substitute the values into the formula and compute the approximate value of the integral.

5. The result obtained from the calculation will be the approximation of the integral $ \int_0^2 \sqrt{4(1+x^2)} \, dx $ using Simpson's rule with $ n = 8 $.