Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Working in polar coordinates, calculate the arc length of cardioid r=1+cosθ.
According to Table 7.2.1, the arc length is obtained by evaluating the integral ∫θ1θ2r2+r′2ⅆθ for r=1+cosθ. The radical in the integrand is easily seen to be
so the complete integration reduces to
2∫02 πcosθ/2 ⅆθ
=2∫0πcosθ/2 ⅆθ−∫π2 πcosθ/2 ⅆθ
=22 sinθ/20π−2 sinθ/2π2 π
Note well that integrating cosθ/2 instead of cosθ/2 would lead to the incorrect answer of zero! An alternative to splitting the integral of the absolute value at θ=π would be to invoke the symmetry in the integrand and reduce the interval of integration to 0,π, with the value of that integral simply being doubled.
Expression palette: Definite Integral template and ordinary derivative template
Fill in fields appropriately.
Context Panel: Evaluate and Display Inline
∫02 π1+cosθ2+ⅆⅆ θ 1+cosθ2ⅆθ = 8
A more elegant interactive solution can be obtained if first, the cardioid r=1+cosθ is defined as a function Rθ. (The variable R is used to avoid assigning to the "working" variable r.)
Context Panel: Assign Function
Rθ=1+cosθ→assign as functionR
Expression palette: Definite Integral template
∫02 πR2θ+R′θ2ⅆθ = 8
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