MediumParametric & polar

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §9.3 Slope, Length, and Area for Polar Curves, #19 · Calculus · Parametric & polar

(b) Compute the length integral

The groove in a record is nearly a spiral $r = c θ$ :
$length = \int r^{2} + (d r / d θ)^{2} d θ = \int_{6}^{14} r^{2} + c^{2} d r / c .$
Take $c = .002$ to give 636 turns between the outer radius 14 cm and the inner radius 6 cm ( $14 - 6$ equals $.002 (636) 2 π$ ).

(a) Omit $c^{2}$ and just integrate $r d r / c$ .

(b) Compute the length integral. Tables and calculators allowed. You will never trust integrals again.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §9.3 Slope, Length, and Area for Polar Curves, #19

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