How do you find the length of a curve using integration?

Question

How  do you find the length of a curve using integration?

Emilia Aguilar · Accepted Answer

If you want to find the arc length of the graph of #y=f(x)# from #x=a# to #x=b#, then it can be found by
#L=int_a^b sqrt{1+[f'(x)]^2}dx#

Isabella Ackerman · Answer

undefined To find the length of a curve using integration, follow these steps:

1. Divide the curve into small segments, each represented by a straight line or a curve.
2. Determine the length of each segment using the distance formula or arc length formula, depending on whether the segment is straight or curved.
3. Sum up the lengths of all segments.
4. Take the limit as the number of segments approaches infinity to get an integral.
5. Set up the integral to represent the length of the curve.
6. Integrate the appropriate expression over the interval of interest to find the length of the curve.

Emilia Aguilar · Answer

undefined To find the length of a curve using integration, you can follow these steps:

1. Define the curve with a function $ y = f(x) $ or $ x = g(y) $, depending on whether it's described in terms of $ x $ or $ y $.
2. Determine the interval over which you want to find the length of the curve.
3. Use the formula for arc length, which is given by:

$$ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$

or

$$ L = \int_c^d \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy $$

depending on whether the curve is described in terms of $ x $ or $ y $, respectively.
4. Integrate the expression within the square root over the given interval.
5. Evaluate the definite integral from the lower limit $ a $ (or $ c $) to the upper limit $ b $ (or $ d $).

This process yields the length of the curve over the specified interval.