   Chapter 10.4, Problem 47E

Chapter
Section
Textbook Problem

Find the exact length of the polar curve.47. r = θ2, 0 ≤ θ ≤ 2π

To determine

To Find: The exact length of the polar curve.

Explanation

Given:

The polar equation with limits tends to be 0θ2π .

r=θ2 (1)

Calculation:

Calculate the value of drdθ .

Differentiate equation (1) with respect to θ .

r=θ2drdθ=2θ

Calculate the exact length of the polar curve using the formula.

L=abr2+(drdθ)2dθ (2)

Substitute θ2 for r and 2θ for drdθ in the equation (2).

L=0π((θ2)2+(2θ)2)dθ=0π(θ4+4θ2)dθ

L=0π(θθ2+4)dθ (3)

Assume the expression for t versus θ as below.

t=θ2+4 (4)

Differentiate the equation (4) with respect to θ .

dtdθ=2θdt=2θdθdt2=θdθ

Substitute 0 for θ in the equation (4).

t=0+4=4

Substitute 2π for θ in the equation (4).

t=(2π)+4=4π2+4

Therefore, integrate the expression within the limits from 4 to 4π2+4

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