Expert answers to "how to find Find the exact length of..."

sonkitty2h

Answered question

2021-12-04

how to find
Find the exact length of the polar curve.

= r = θ^{2}, 0 \leq θ \leq 7 \frac{π}{4}

rerCessbalmuh

Beginner2021-12-05Added 13 answers

Step 1
The polar curve i given as,

r = θ^{2}, 0 \leq θ \leq 7 \frac{π}{4}

The formule to calculate the exact length of the polar curve is,

= \int_{a}^{b} (r^{2} + \frac{d r}{d θ}) d θ

......(1)
Step 2
Compute

= d \frac{r}{d} θ

= \frac{d r}{d θ} = \frac{d}{d θ} (θ^{2})

= 2 θ

Step 3
Substitute

r = θ^{2}, d \frac{r}{d} θ = 2 θ a = 0

and

b = 7 \frac{π}{4}

in equation (1).

l = \int_{0}^{7 \frac{π}{4}} (θ^{2} + 2 θ) d θ

= \int_{0}^{7 \frac{π}{4}} (θ^{2}) d θ + \int_{0}^{7 \frac{π}{4}} (2 θ) d θ

= {[\frac{θ^{3}}{3}]}_{0}^{7 \frac{π}{4}} + {[\frac{2 θ^{2}}{2}]}_{0}^{7 \frac{π}{4}}

= \frac{1}{3} {(\frac{7 π}{4})}^{3} + {(\frac{7 π}{4})}^{2}

= \frac{49 π^{2}}{16} (\frac{7 π}{12} + 1)

= \frac{49 π^{2}}{16} (\frac{7 π + 12}{12})

= \frac{{49 π^{2}} ({7 π + 12})}{192}

Step 4
The exact length of polar curve is,

= \frac{{49 π^{2}} ({7 π + 12})}{192}

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