Question

In Exercises 13-18, (a) use Simpson's Rule with n = 4 to approximate the value of the integral and (b) find the exact value of the integral to check your answer. (Note that these are the same integrals as Exercises 1-6, so you can also compare it with the Trapezoidal Rule approximation.) $$\int_{0}^{\pi} \sin x d x$$

Short answer

The short answer will be two values: the integral value calculated using the Simpson's rule and the integral value calculated exactly by actually performing the integration. These values will answer parts (a) and (b) of the problem respectively.

Step-by-step solution

Apply Simpson's Rule

To apply Simpson's Rule, we'll be using the formula: \[ S = \frac{h}{3} [f(a) + 4\sum_{i=1}^{(n-1)/2}f(a + 2i*h) + 2\sum_{i=1}^{(n-2)/2}f(a + (2i+1)*h) + f(b)] \] where \(a\) and \(b\) are the limits of integration, \(n\) is the number of intervals and \(h = (b - a) / n\). Now, substituting \(a = 0\), \(b = \pi\), \(n = 4\), \(h = \frac{\pi - 0}{4} = \frac{\pi}{4}\) and \(f(x) = \sin{x}\), we can compute the integral estimation.

Calculate the exact value

Now we need to calculate the exact value of the integral for comparing purpose. For that, we will perform the integration of \(\sin x\) from 0 to \(\pi\). The antiderivative of \(\sin x\) is \(-\cos x\), so we substitute the limits into \( -\cos x\).

Compare approximate value with exact value

In this final step, we compare the value obtained using Simpson's Rule with the exact value. This will give us an idea of how close (or far) our approximation is to the actual value.