Question

Determine the arc length of the curve.

Short answer

The resultant answer is \({S_{10}} = 2.280731\), with the help of the calculator where \(L = 2.280526\).

Step-by-step solution

Given data

The curve function is \(y = {e^{ - {x^2}}}\).

The lower limit is \(a = 0\) and the upper limit is \(b = 2\).

The number of intervals is 10.

Concept of Arc length

Arc length is the distance between two points along a section of a curve.

Determine the length of the curve

The length of the curve is given by:

\(\begin{aligned}{}L = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} } dx\\L = \int_0^2 {\sqrt {1 + {{\left( {\frac{{{{\left. {d\left( {{e^{ - {x^2}}}} \right)} \right)}^2}}}{{dx}}} \right)}^2}} } dx\\L = \int_0^2 {\sqrt {1 + \left( { - 2x{e^{{{\left. { - {x^2}} \right)}^2}}}} \right.} } dx\\L = \int_0^2 {\sqrt {1 + 4{x^2}{e^{ - 2{x^2}}}} } dx\end{aligned}\)

Simplify further: \(L = \int_0^2 f (x)dx\), where \(f(x) = \sqrt {1 + 4{x^2}{e^{ - 2{x^2}}}} \).

Simplify and substitute the values

By Simpson's rule:

\(\begin{aligned}{}\Delta x = \frac{{2 - 0}}{{10}} = 0.2\\{S_{10}} = \frac{{\Delta x}}{3}(f(0) + 4f(0.2) + 2f(0.4) + 4f(0.6) + 2f(0.8) + 4f(1) + 2f(1.2) + 4f(1.4) + \\2f(1.6) + 4f(1.8) + f(2))\end{aligned}\)

Substitute \(\Delta x = 0.2\) and use the functional values from the table.

\(\begin{aligned}{}{S_{10}} = \frac{{0.2}}{3}(1 + 4(1.071307) + 2(1.2102625) + 4(1.3041945) + 2(1.3083484) + 4(1.2415076)\\ + 2(1.1503635) + 4(1.0749671) + 2(1.0301429) + 4(1.0098902) + (1.0052697))\\{S_{10}} = 2.280731\end{aligned}\)

Since, \(L = \int_0^2 {\sqrt {1 + 4{x^2}{e^{ - 2{x^2}}}} } dx \approx 2.280526\), this is very close to the Simpson’s approximation.