Question

What is the arc length of the polar curve `f(theta) = 5sintheta-4theta` over `theta in [pi/8, pi/3]`?

Answer 1

L `approx` 0.47235219096 `approx` 0.472 (3 decimal places)

Explanation:

The derivation of the arc length of a polar curve over the interval `[a, b]` could be found in another one of my posts, the link to which is here: https://socratic.org/questions/what-is-the-arclength-of-the-polar-curve-f-theta-3sin-3theta-2cot4theta-over-the#455941

Applying to this question, we must first find the derivative of the function with respect to `theta`:

`f'(theta) = d/(d theta)(5 sin(theta) - 4 theta)`

Since `d/(d theta) (sin (theta)) = cos(theta)`,

`f'(theta) = 5cos(theta) - 4`

The formula for the arc length of a polar curve over the interval `[a, b]` is given by:

`L = int_a^bsqrt(r^2 + ((dr)/(d theta))^2) d theta`

where `r = f(theta)` and therefore it follows that `(dr)/(d theta) = f'(theta)`

Therefore, the arc length for the given function over the given interval would be:

`L = int_(pi/8)^(pi/3)sqrt((5 sin(theta) - 4 theta)^2 +( 5cos(theta) - 4)^2) d theta`

Using a graphing utility (for the sake of simplicity), the integral equals about `0.47235219096`, or to three decimal places, it is equal to `0.472`