Length of a curve

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)

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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!