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1. The problem statement, all variables and given/known data
Use Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi##, and then determine the exact length of the curve and a four decimal approximation.
Hint: ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left\sec(x)+\tan(x)\right+C##
2. Relevant equations
Arc Length of a Polar Curve is given as:
$$\begin{align} S & = \int_\alpha^{\beta}\sqrt{\left[f(\theta)\right]^2+\left[f'(\theta)\right]^2}d\theta\\
& = \int_\alpha^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta \end{align}$$
3. The attempt at a solution
I have attached the required graph from Desmos.
Right off the bat, I am stuck here.
I start off with ##S=\int_0^{4\pi}\sqrt{(\theta)^2 + (\theta')^2}## and this makes no sense.
I also tried using the identity ##r^2=x^2+y^2## but then it of course gave me a nasty polynomial in two variables as the integrand.
I'm not seeing where this hint is coming into play.
Any ideas?
Use Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi##, and then determine the exact length of the curve and a four decimal approximation.
Hint: ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left\sec(x)+\tan(x)\right+C##
2. Relevant equations
Arc Length of a Polar Curve is given as:
$$\begin{align} S & = \int_\alpha^{\beta}\sqrt{\left[f(\theta)\right]^2+\left[f'(\theta)\right]^2}d\theta\\
& = \int_\alpha^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta \end{align}$$
3. The attempt at a solution
I have attached the required graph from Desmos.
Right off the bat, I am stuck here.
I start off with ##S=\int_0^{4\pi}\sqrt{(\theta)^2 + (\theta')^2}## and this makes no sense.
I also tried using the identity ##r^2=x^2+y^2## but then it of course gave me a nasty polynomial in two variables as the integrand.
I'm not seeing where this hint is coming into play.
Any ideas?
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