Home > Calculus > Determining the Length of a Curve

What is the arclength of #f(x)=(x-2)/x^2# on #x in [-2,-1]#?

Answer 1

# f(x) = (x-2)/x^2 #

# :. f'(x) = ( (x^2)(1) - (x-2)(2x) ) / (x^2)^2 # # :. f'(x) = ( x^2 - 2x^2 +4x ) / x^4 # # :. f'(x) = ( 4x-x^2 ) / x^4 # # :. f'(x) = ( 4-x ) / x^3 #

Arc Length is given by: # L = int_(-2)^(-1) sqrt (1+f'(x)^2) dx # # :. L = int_(-2)^(-1) sqrt (1+(( 4-x ) / x^3)^2) dx # # :. L = int_(-2)^(-1) sqrt (1+( 4-x )^2 / x^6) dx # # :. L = int_(-2)^(-1) sqrt ((x^6+( 4-x )^2) / x^6) dx # # :. L = int_(-2)^(-1) sqrt (x^6+( 4-x )^2) / x^3 dx #

This definite integral does not have an elementary intrinsic solution and would need to be solved numerically, using either a computer or estimated using the Trapezium Rule or Simpson's Rule

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Agard

To find the arc length of the function ( f(x) = \frac{x - 2}{x^2} ) on the interval ([-2, -1]), you can use the arc length formula:

[ L = \int_{-2}^{-1} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

where ( \frac{dy}{dx} ) represents the derivative of ( f(x) ) with respect to ( x ). Calculate ( \frac{dy}{dx} ), then substitute it into the arc length formula and integrate over the given interval to find the arc length ( L ).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.