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