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.
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