Question

Consider the integral \(\int_{0}^{\pi} \sin x d x\) (a) Use a calculator program to find the Trapezoidal Rule approximations for n = 10, 100, and 1000. (b) Record the errors with as many decimal places of accuracy as you can. (c) What pattern do you see? (d) Writing to Learn Explain how the error bound for \(E_{T}\) accounts for the pattern.

Short answer

The integral of sin(x) from 0 to pi is 2. This was calculated analytically and then trapezoidal rule was applied for n=10, 100, and 1000. Approximations were obtained and differences were calculated as errors. A pattern was observed in these errors: it decreases as 'n' goes up. This pattern is due to the reduced error boundary based on the trapezoidal rule, as increasing 'n' leads to a better approximation, reducing the error.

Step-by-step solution

Analytical solution

Go ahead and find the analytical integral from 0 to pi of sin(x) dx, which equals to -cos(x) evaluated from 0 to pi and is equal to 2.

Approximation using trapezoidal rule

Now use the trapezoidal rule to approximate the integral. First for n=10, then for n=100 and finally for n=1000. The integral approximations can be computed using available software or by hand if you are familiar with the trapezoidal rule formula.

Compute the errors

Subtract each of these approximations from the actual value (2) to find the errors. Record these errors with as much precision as possible.

Pattern observation

Observe the pattern of these error values as n increases. It is expected that the approximation gets closer to the real value, decreasing the error.

Explain the error bound for \(E_{T}\)

Based on the pattern observed, explain how the trapezoidal rule's error bound accounts for this pattern. It is expected that as the number of trapezoids (n) increases, the approximation becomes more accurate, thus the error decreases.