Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Find the exact value of the integral.

Question

Quizplus · Accepted Answer

Using the trapezoidal rule with $n=4$, we get:

$$\begin{aligned} \int_{-1}^1e^x\ dx &\approx \frac{1}{2}\left[f(-1)+2f\left(-\frac{1}{2}ight)+2f\left(\frac{1}{2}ight)+f(1)ight]\ &= \frac{1}{2}\left[e^{-1}+2e^{-1/2}+2e^{1/2}+eight]\ &= 2.233665\end{aligned}$$

Using Simpson's rule with $n=4$, we get:

$$\begin{aligned} \int_{-1}^1e^x\ dx &\approx \frac{1}{3}\left[f(-1)+4f\left(-\frac{1}{2}ight)+2f(0)+4f\left(\frac{1}{2}ight)+f(1)ight]\ &= \frac{1}{3}\left[e^{-1}+4e^{-1/2}+2+4e^{1/2}+eight]\ &= 2.350402\end{aligned}$$

The exact value of the integral is:

$$\begin{aligned} \int_{-1}^1e^x\ dx = e- \frac{1}{e} = 2.350402\end{aligned}$$

Therefore, the best choice is $\boxed{	extbf{(C)}}$.

Use the Trapezoidal Rule and Simpson's Rule to Approximate the Value