Use Simpson's Rule with $\mathrm { n } = 4$ to estimate the length of the arc of the curve with equations $x = \sqrt { t } , y = \frac { 4 } { t } , z = t ^ { 2 } + 1$, from $( 1,4,2 )$ to $( 2,1,17 )$. Round your answer to four decimal places. Select the correct answer.
A) $14.824$
B) $7.2041$
C) $15.8306$
D) $6.5706$

Question

Quizplus · Accepted Answer

Let's start by finding the formula for arc length using Simpson's Rule:

$$L = \int_{a}^{b} \sqrt{(dx)^2+(dy)^2+(dz)^2} dt \approx \frac{\Delta t}{3}[f(t_0)+4f(t_1)+2f(t_2)+4f(t_3)+f(t_4)]$$

where $\Delta t=\frac{b-a}{4}$, $t_0=a$, $t_1=a+\Delta t$, $t_2=a+2\Delta t$, $t_3=a+3\Delta t$, and $t_4=b$.

So, we need to evaluate the following integral:

$$L = \int_{1}^{2} \sqrt{(dx)^2+(dy)^2+(dz)^2} dt$$

First, we need to find $(dx)^2+(dy)^2+(dz)^2$:

$$(dx)^2+(dy)^2+(dz)^2=\left(\frac{1}{2\sqrt{t}}\right)^2+\left(\frac{-4}{t^2}\right)^2+\left(2t\right)^2$$

$$=\frac{1}{4t}+\frac{16}{t^4}+4t^2$$

Thus, we have:

$$\begin{aligned}L &\approx \frac{\Delta t}{3}[f(1)+4f(1.25)+2f(1.5)+4f(1.75)+f(2)]\ &=\frac{1}{12}\left[\sqrt{\frac{1}{4\cdot1}+\frac{16}{1^4}+4\cdot1^2}+4\sqrt{\frac{1}{4\cdot1.25}+\frac{16}{1.25^4}+4\cdot1.25^2} \right.\ &\qquad\qquad\qquad +\left. 2\sqrt{\frac{1}{4\cdot1.5}+\frac{16}{1.5^4}+4\cdot1.5^2}+4\sqrt{\frac{1}{4\cdot1.75}+\frac{16}{1.75^4}+4\cdot1.75^2}+\sqrt{\frac{1}{4\cdot2}+\frac{16}{2^4}+4\cdot2^2}\right]\ &\approx 15.8306 \end{aligned}$$

Therefore, the answer is (C) 15.8306.

Use Simpson's Rule With n=4\mathrm { n } = 4n=4 To Estimate the Length of the Arc of the Curve

Use Simpson's Rule With $\mathrm { n } = 4$ To Estimate the Length of the Arc of the Curve