Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Working in polar coordinates, calculate the complete arc length of the limaçon r=1/2−cosθ.
According to Table 7.2.1, the arc length is obtained by evaluating the integral ∫θ1θ2r2+r′2ⅆθ for r=1/2− cosθ. The radical in the integrand is easily seen to be
so the complete integration reduces to
12∫02 π5−4 cosθⅆθ
The antiderivative of 5−4 cosθ is not an elementary function; it is a member of the family of elliptic functions that Maple knows. Without Maple, this integral could only be evaluated numerically, or with a table of integrals.
An elegant interactive solution is possible if the limaçon is defined as a function Rθ.
Context Panel: Assign Function
Rθ=1/2−cosθ→assign as functionR
Expression palette: Definite Integral template
Press the Enter key
Context Panel: Approximate≻10 (digits)
→at 10 digits
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