Question

Use Simpson Rule with \(n = 6\)to estimate the length of curve

\(x = t - {e^t},y = t + {e^t}, - 6 \le t \le 6\)

Short answer

Length of the Curve is \( \approx 612.3053\)..

Step-by-step solution

Step-1(Using Integration to represent length of Curve)

The formula for length of curve is

\(L = \int\limits_\alpha ^\beta {\sqrt {{{(\frac{{dx}}{{dt}})}^2} + {{(\frac{{dy}}{{dt}})}^2}} } .dt\)

Where, \(\alpha \)and \(\beta \) represent lower limit and upper limit of t respectively…

On Differentiating x and y w.r.t \(t\).We get,

\(\frac{{dx}}{{dt}} = 1 - {e^t}\)and \(\frac{{dy}}{{dt}} = 1 + {e^t}\)

Step-2(Calculation of length of curve)

\({(\frac{{dx}}{{dt}})^2} + {(\frac{{dy}}{{dt}})^2} = \)\({(1 - {e^t})^2} + {(1 + {e^t})^2}\)

\(1 + {e^{2t}} - 2{e^t} + 1 + {e^{2t}} + 2{e^t}\)

\( = 2 + 2{e^{2t}}\)

By Using the formula,

\(L = \int\limits_\alpha ^\beta {\sqrt {{{(\frac{{dx}}{{dt}})}^2} + {{(\frac{{dy}}{{dt}})}^2}} } .dt\)

\( = \int\limits_{ - 6}^6 {\sqrt {2 + 2{e^{2t}}} } dt\)\(\)

Now,using Simpson rule for approximation

\(f(t) = \sqrt {2 + 2{e^{2t}}} \)

For \(n = 6,\Delta t = \frac{{6 + 6}}{6} = 2\)

Sub intervals are,

\(\)\(( - 6, - 4),( - 4, - 2),( - 2,0),(0,2),(2,4),(4,6)\)

By Simpson Rule

\(L \approx \frac{{\Delta t}}{t}(f( - 6)) + 4f( - 4)) + (2f( - 2)) + (4f(0)) + (2f(2)) + (4f(4)) + f(6))\)

\( \approx \frac{2}{3}(\sqrt {2 + 2{e^{ - 12}}} + 4\sqrt {2 + 2{e^{ - 8}}} + 2\sqrt {2 + 2{e^{ - 4}}} + 4\sqrt 4 + 2\sqrt {2 + 2{e^4}} + 4\sqrt {2 + 2{e^4}} + \sqrt {2 + 2{e^{12}}} )\)

\(L \approx 612.3053\)

\(\)Hence The Length of Given Parametric Curve is \( \approx 612.3053\) unit.