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

1 Answer
Jul 30, 2017

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