Find the arc length function for the curve y = 2x^3/2 with starting point P0(36, 432).

### 1 Answer

It the length starts at one point and never ends , it is infinite length from P0 (36,432) in which direction . Do you want the distance to a random point P2( x_2, y_2) the formula given in Calculus textbook https://www.math.hmc.edu/calculus/tutorials/arc_le… L = ∫ √ ̅( 1 + ( f'(x))^2 ) dx from x = a to x= b f'(x) = (3/4) x^(1/2) L = ∫ √ ̅ (1+ (3/4)x^(1/2))^2) dx = L = ∫ √ ̅ (1 + (9/16) x ) dx = see this reference and how to find the derivative of the above https://www.symbolab.com/solver/implicit-derivativ… L = (1/54) ( 16 +9x )^(3/2) from point a to b so if point P0 is (36, 432) P1 ( x_2, y_2) L = (1/54) (16 + 9*x_2)^(3/2) – (1/54) (16 + 9 *36)^(3/2) L = (1/54) (16 + 9*x_2)^(3/2) – (1/54) (340)^(3/2)

L = (1/54) (16 + 9*x_2)^(3/2) – approx. 116.0979672