Chapter 5: Applications of Integration
Section 5.4: Arc Length
Example 5.4.3
Obtain the length of the curve defined by y=sinx,0≤x≤ π.
Solution
Mathematical Solution
Application of the appropriate member of Table 5.4.1 leads to
∫0π1+ⅆⅆ x sinx2ⅆx
= ∫0π1+cos2xⅆx
=22EllipticE122
≐3.820197788
The antiderivative of 1+cos2x cannot be found in terms of elementary functions. It is expressed in terms of EllipticEz,k, the incomplete elliptic integral, and EllipticEk, the complete elliptic integral. The point here is not to have the reader delve into the definitions of these special functions, but rather, to have the reader understand that arc-length integrals are amongst the most difficult integrals to be encountered in a calculus course. If the power of Maple is not to be used to evaluate these integrals, then very few such integrals can be evaluated with just the limited tools of hand-calculations.
Maple Solution
Figure 5.4.3(a) shows the tutor applied to finding the arc length of sinx on the interval 0,π.
In the graph, the function is in red; the integrand, in blue; and the arc-length function, in green.
The arc-length integral is evaluated in terms of the complete elliptic function EllipticEk, where k=1/2. The floating-point equivalent of the exact value is the last line of the display.
The ArcLength command in the Student Calculus1 package will output the arc length, or the graph displayed in the tutor.
Figure 5.4.3(a) Arc Length tutor
The ArcLength command
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Apply the ArcLength command with the output option set to integral.
Press the Enter key to obtain the unevaluated arc-length integral.
ArcLengthsinx,0..π,output=integral
∫0π1+cosx2ⅆx
Apply the ArcLength command. Press the Enter key.
Context Panel: Approximate≻10 (digits)
ArcLengthsinx,0..π
22EllipticE122
→at 10 digits
3.820197788
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