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: | + | 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 = <font size = "+2"><span>∫</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]) | |
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| − | length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx (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] | ||
| + | ------- | ||
For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup> | For the previous example of heating a liquid, the first derivative of the equation was f '(t)=30e<sup>−0.3t</sup> | ||
Latest revision as of 18:12, 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!
