Question

Find the exact length of the polar curve. r = θ2, 0 ≤ θ ≤ 7π/4

Answer 1

`\displaystyle{L=((49\pi^2+64)√(49\pi^2+64)-512)/(192)}`

Explanation:

Length of polar curve has formula of:

`\displaystyle{L = \int_( \theta_1 )^( \theta_2) \sqrt{ {r}^(2) + \left( (dr)/(d \theta) \right)^2 } \: d \theta}`

From power rule, we know that:

`\displaystyle{ (dr)/(d \theta) = 2 \theta}`

Therefore, substitute r, angles and its derivative in:

`\displaystyle{L = \int_(0 )^{ (7\pi)/(4) }\sqrt{ { \theta}^(4) + \left( 2 \theta \right)^2 } \: d \theta} \\ \\ \displaystyle{L = \int_(0 )^{ (7\pi)/(4) }\sqrt{ { \theta}^(4) + 4 \theta^(2) } \: d \theta} \\ \\ \displaystyle{L = \int_(0 )^{ (7\pi)/(4) }√( \theta^2(\theta^2+4)) \: d \theta} \\ \\ \displaystyle{L = \int_(0 )^{ (7\pi)/(4) } \theta √( \theta^2+4)\: d \theta}`

Let the following:

`\displaystyle{u=√(\theta^2+4)\to u^2=\theta^2 + 4 \to 2u \: du = 2\theta \: d\theta \to u\: du=\theta \: d\theta}`

Therefore, substitute in appropriate terms:

`\displaystyle{L = \int_(0 )^{ (7\pi)/(4) } u √( u^2)\: d u}\\\\\displaystyle{L = \int_(0 )^{ (7\pi)/(4) } u ^2\: d u}`

Since we now use u-term as the interval, we have to substitute theta intervals in
`\displaystyle{u=√(\theta^2+4)}` to find new intervals. Hence:

`\displaystyle{u_(\theta_2)=\sqrt{\left((7\pi)/(4)\right)^2+4} =\sqrt{(49\pi^2)/(16)+4} =\sqrt{(49\pi^2)/(16)+(64)/(16)} =\sqrt{(49\pi^2+64)/(16)}}\\\\\displaystyle{u_(\theta_1)=√(0^2+4)=√(4)=2}`

Now we have our new interval of:

`\displaystyle{L = \int_(2)^{ \sqrt{(49\pi^2+64)/(16)} } u ^2\: d u}`

Then we can apply definite integration rule:

`\displaystyle{L = \left[ (u^3)/(3)\right]_(2)^{ \sqrt{(49\pi^2+64)/(16)} } }\\\\\displaystyle{L= \frac{\left(\sqrt{(49\pi^2+64)/(16)\right)^3} }{ 3} - (2^3)/(3)}\\\\\displaystyle{L=\frac{(49\pi^2+64)/(16)\sqrt{(49\pi^2+64)/(16)}}{3}-(8)/(3)}\\\\\displaystyle{L=((49\pi^2+64)/(64)√(49\pi^2+64))/(3)-(8)/(3)}\\\\\displaystyle{L=((49\pi^2+64)√(49\pi^2+64))/(192)-(8)/(3)}`

`\displaystyle{L=((49\pi^2+64)√(49\pi^2+64))/(192)-(512)/(192)}\\\\\displaystyle{L=((49\pi^2+64)√(49\pi^2+64)-512)/(192)}`

And that’s the answer!