How do you Use Simpson's rule to approximate the integral #int_0^1f(x)dx# with #n = 10#?

Question

Christopher Agresta · Accepted Answer

Simpsn's Rule with #n=10# looks like this:
#S_{10}={Delta x}/{3}[f(0)+4f(0.1)+2f(0.2)+4f(0.3)+2f(0.4)+4f(0.5)+2f(0.6)+4f(0.7)+2f(0.8)+4f(0.9)+f(1)]#,
where #Delta x={b-a}/n={1-0}/{10}=0.1#. Would this get you going?

Paisley Johnson · Answer

undefined To use Simpson's rule to approximate the integral $ \int_{0}^{1} f(x) \, dx $ with $ n = 10 $, where $ n $ is the number of subintervals, follow these steps:

1. Divide the interval $ [0, 1] $ into $ n $ subintervals. Since $ n = 10 $, each subinterval will have width $ h = \frac{1 - 0}{10} = 0.1 $.

2. Compute the values of the function $ f(x) $ at the endpoints of each subinterval and the midpoint. Let's denote these values as $ f(x_i) $, $ f(x_{i-1}) $, and $ f(\frac{x_i + x_{i-1}}{2}) $, where $ x_i $ and $ x_{i-1} $ are the endpoints of the subinterval.

3. Apply Simpson's rule formula for each subinterval:

$$ \text{Approximation for each subinterval} = \frac{h}{6} [f(x_{i-1}) + 4f(\frac{x_i + x_{i-1}}{2}) + f(x_i)] $$

4. Sum up the approximations for all subintervals to get the final approximation for the integral:

$$ \text{Approximation for the integral} = \sum_{i=1}^{10} \frac{h}{6} [f(x_{i-1}) + 4f(\frac{x_i + x_{i-1}}{2}) + f(x_i)] $$

Substitute the appropriate values of $ x_i $, $ x_{i-1} $, and $ h $ into the formula to compute the approximation.

Adam Adkins · Answer

undefined To use Simpson's rule to approximate the integral $ \int_{0}^{1} f(x) \, dx $ with $ n = 10 $, follow these steps:

1. Divide the interval $[0, 1]$ into $ n = 10 $ subintervals of equal width. Since $ n $ is even, this will result in $ n/2 = 5 $ intervals.
2. Determine the step size $ h $ using the formula $ h = \frac{b - a}{n} $, where $ a = 0 $ and $ b = 1 $.
3. Evaluate the function $ f(x) $ at the endpoints of each subinterval and at the midpoint of every two consecutive subintervals. This will give you $ f(x_0), f(x_1), f(x_2), \ldots, f(x_n) $, where $ x_i = a + ih $ for $ i = 0, 1, 2, \ldots, n $.
4. Apply Simpson's rule formula:
   $$ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] $$
5. Substitute the values of $ f(x_i) $ into the formula and perform the calculations to approximate the integral.

By following these steps, you can use Simpson's rule to approximate the integral $ \int_{0}^{1} f(x) \, dx $ with $ n = 10 $.