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 = f(x) | + | equation of curve = f(x) |
first derivative = f '(x) | first derivative = f '(x) | ||
| − | length = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx | + | length of curve = <font size = "+2"><span>∫</span></font>sqr(1+f '(x)<sup>2</sup>)dx |
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> | ||
Revision as of 18:09, 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
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!
