Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Starting with the expression for the arc length of a curve defined parametrically, obtain the expression for the arc length of a curve defined by r=rθ in polar coordinates. (See Table 7.2.1.)
From Table 5.4.1, the arc length of the curve defined parametrically by the equations x=xt,y=yt is given by the integral
The equations x=rθcosθ,y=rθsinθ convert the polar curve to Cartesian coordinates with θ now the parameter. Hence, the radicand in the integrand becomes
=r′cosθ−r sinθ2+r′sinθ+r cosθ2
= r′2cos2θ−2 rr′cosθsinθ+r2sin2θ +r′2sin2θ+2 rr′cosθsinθ+r2cos2θ
from which it follows that
as per Table 7.2.1.
Define the parametric equations x=xθ and y=yθ
Context Panel: Assign Function
xθ=rθcosθ→assign as functionx
yθ=rθsinθ→assign as functiony
Obtain the radicand of the integrand for the arc-length integral
Write the sum of the squares of the derivatives.
Press the Enter key.
Context Panel: Simplify≻Simplify
The expression for L in Table 7.2.1 is now easily obtained.
<< 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)