EEristavi said:Homework Statement
I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1)
Homework Equations
s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b
The Attempt at a Solution
I'm bit confused:
I'm thinking of writing function regarding x, f(x). However, I can also write for y, f(y).
Which is better? and what values should I take for definite integral ("limits")
Dick said:What does that tell you about limits on xxx?
EEristavi said:well, from the problem statement x should change from 0 to 1, but its not correct (as I see from the solution), integral is taken from 1 to 2 (and I can't figure it out why)
Dick said:You do know that, e.g., at the point (1,0)(1,0)(1,0) the value of xxx is 1, right?
EEristavi said:OK, now I think, I need more clarification... I will write what is in my mind:
point (1, 0) means that y=1, x=0
but I see from function y = f (x), that if x = 0 => y = 0
So I guess, here is my problem of understanding...
Ok, maybe it's been a while, since I've done some mathematics :DDick said:Where I come from the point ##(1,0)## means ##x=1## and ##y=0##. That looks like the definition the problem is using also. I don't think I've ever heard of a different convention.
The formula for finding the length of a curve is known as the arc length formula, which is given by:
L = ∫√(1 + (dy/dx)^2) dx, where dy/dx is the derivative of the curve's equation.
To find the derivative of a curve's equation, you can use the power rule, chain rule, or product rule depending on the complexity of the equation. In this case, the derivative of y2 = (x-1)3 is 3(x-1)2.
No, the arc length formula involves integration, which is a fundamental concept in calculus. Therefore, it is not possible to find the length of a curve without using calculus.
This equation represents a cubic curve that is shifted one unit to the right. It can be used to model various real-life situations, such as the growth of a population or the trajectory of a projectile.
Yes, the arc length formula can only be used to find the length of a curve with a continuous and differentiable equation. It also requires the use of calculus, which may not be suitable for all situations.