Difference between revisions of "Length of a curve"

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The length of a curve from point a to point b can be found using an integral of the first derivative of the equation:
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The length of a curve from point a to point b can be found using an integral of the first derivative of the equation: equation of curve  = f(x), first derivative = f '(x)
 
 
equation of curve  = f(x)
 
 
 
first derivative = f '(x)
 
  
 
length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx  (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])
 
length of curve = <font size = "+2"><span>&#8747;</span></font>sqr(1+f '(x)<sup>2</sup>)dx  (this equation can be derived using the method [https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al)/6%3A_Using_Definite_Integrals/6.1%3A_Using_Definite_Integrals_to_Find_Area_and_Length#:~:text=In%20addition%20to%20being%20able%20to%20use%20definite,small%20pieces%20whose%20lengths%20we%20can%20easily%20approximate. here])

Revision as of 18:11, 1 April 2021

The length of a curve from point a to point b can be found using an integral of the first derivative of the equation: equation of curve = f(x), first derivative = f '(x)

length of curve = sqr(1+f '(x)2)dx (this equation can be derived using the method here)

For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e−0.3t

so the length of the curve from 0 to 5 minutes would be:

sqr(1+(30e−0.3t)2)dt

entering this equation into the integral calculator here gives 77.8 degrees, which is about the same value calculated in the previous example using integration of the rate of change!