Use Simpson's Rule with $n = 6$ to estimate the length of the curve $y = 3 \sin x , 0 \leq x \leq \pi$. Round your answer to six decimal places.
A) $6.987208$
B) $6.947368$
C) $6.972089$
D) $6.947582$
E) None of these

Question

Quizplus · Accepted Answer

Using Simpson's Rule with $n = 6$ gives:

$$\begin{aligned} \text{Length} &= \frac{\pi-0}{3n}\left[y_0 + 4y_1+ 2y_2+ 4y_3+ 2y_4+ 4y_5+ y_6\right]\ &= \frac{\pi}{18}\left[ y_0 + 4y_1+ 2y_2+ 4y_3+ 2y_4+ 4y_5+ y_6 \right]\ &= \frac{\pi}{18}\left[ y_0 + 4y_1+ 2y_2+ 4y_3+ 2y_4+ 4y_5+ y_6 \right]\ &= \frac{\pi}{18}\left[ 3\sin(0) + 4\cdot3\sin\left(\frac{\pi}{6}\right) + 2\cdot3\sin\left(\frac{\pi}{3}\right) + 4\cdot3\sin\left(\frac{\pi}{2}\right) + 2\cdot3\sin\left(\frac{2\pi}{3}\right) + 4\cdot3\sin\left(\frac{5\pi}{6}\right) + 3\sin(\pi) \right]\ &= \frac{\pi}{18}\left[ 6 + 6\sqrt{3} + 6 + 12 + 6\sqrt{3} + 6 - 6 \right]\ &= \pi\sqrt{3} \approx 6.947582\end{aligned}$$

Therefore, the answer is D.

Use Simpson's Rule With n=6n = 6n=6 To Estimate the Length of the Curve

Use Simpson's Rule With $n = 6$ To Estimate the Length of the Curve