Finding the length of a polar curve between two points

Question

So the curve r=5(theta)^2 crosses the x axis an infinite amount of times. i have to find the length of the curve between the 1st and 3rd crossing point. the 1st crossing point is theta = 0, and i have calculated the third to be 3pi/2 (not sure if correct). then i have used the equation for solving, by using x= cross(theta) and y= rsin(theta). Then by deriving these two equations, before finding the dy/dx, (i use dy/dtheta and dtheta/dx), however the answer i get ((5/24)*(((16+4(pi^2))^(3/2))-64)) is wrong. anyone know how else to solve this?

Answer 1

You are correct in that you must first parameterize the polar curve. Once you have your parameterization, integrate using the Pythagorean Theorem to get the Arc Length $$AL = \int_a^b{\sqrt{(1 + \left(\frac{dy}{dt}\right)^2} dt}$$

If you are not getting the correct answer I would first verify that you have paramaterized your curve correctly - also write in LaTex next time.