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.
A) $14.824$
B) $7.2041$
C) $15.8306$
D) $6.5706$
E) None of these

Question

Quizplus · Accepted Answer

We need to use Simpson's Rule with $\mathrm { n } = 4$ to evaluate the following integral:
$$\int_{1}^{2}\sqrt{\frac{dx}{dt}}\sqrt{\left(\frac{\partial x}{\partial t}\right)^2+\left(\frac{\partial y}{\partial t}\right)^2+\left(\frac{\partial z}{\partial t}\right)^2}dt$$
Substituting, we have:
$$\int_{1}^{2}\sqrt{\frac{1}{2t}\cdot\left(\frac{1}{2\sqrt{t}}\right)^2+\left(\frac{-4}{t^2}\right)^2+\left(2t\right)^2}dt$$
Simplifying this integral and using Simpson's Rule, we get the following approximation:
$$L \approx \frac{1}{3}\cdot\left[\left(\sqrt{\frac{1}{2}+\frac{16}{16}+5}\right)+4\cdot\left(\sqrt{\frac{1}{5}+\frac{16}{81}+17}\right)+2\cdot\left(\sqrt{\frac{1}{8}+\frac{16}{256}+65}\right)+4\cdot\left(\sqrt{\frac{1}{10}+\frac{16}{625}+257}\right)+\left(\sqrt{\frac{1}{2}+\frac{16}{1024}+513}\right)\right]$$
Simplifying this expression, we get $L \approx 15.8306$, which is closest to option C.

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