Home > Calculus > Determining the Length of a Curve

What is the arc length of #f(x) = x-xe^(x) # on #x in [ 2,4] #?

Answer 1

Christopher Agresta

#L=int_2^4sqrt(1+(1-e^x-xe^x)^2)dxapprox201.361#

The arc length of #f# on #x in [a,b]# is given by

#L=int_a^bsqrt(1+(f'(x))^2)dx#

Here, #f(x)=x-xe^x# so #f'(x)=1-e^x-xe^x#. Then,

#L=int_2^4sqrt(1+(1-e^x-xe^x)^2)dxapprox201.361#

Which can be found with a calculator or Wolfram Alpha.

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

Answer 2

Charles Arocha

To find the arc length of the function ( f(x) = x - xe^x ) on the interval ([2,4]), you can use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + (f'(x))^2} , dx ]

Where ( f'(x) ) is the derivative of the function ( f(x) ).

First, find ( f'(x) ), then substitute it into the formula and integrate over the interval ([2,4]).

[ f'(x) = 1 - (1+x)e^x ]

[ L = \int_{2}^{4} \sqrt{1 + (1 - (1+x)e^x)^2} , dx ]

[ L \approx 4.732 ]

So, the arc length of ( f(x) = x - xe^x ) on the interval ([2,4]) is approximately ( 4.732 ) units.

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.