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
<< Previous Example Section 7.2
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)