method one
In advanced mathematics, we can use derivatives to find the arc length of a curve without using parametric equations. This method is known as arc length differentiation.
Suppose there is a curve whose equation is y = f(x) , and we want the arc length between two points A(x1, y1) and B(x2, y2) on the curve.
First, we denote two points A and B on the curve as x1 and x2 respectively, and then calculate the arc length differential ds of the curve between these two points.
According to the Pythagorean theorem, we can get the expression of ds:
ds = √(dx^2 + dy^2)
where dx = x2 - x1, dy = f(x2) - f(x1).
We can then use the definition of the derivative to approximate dy.
dy = f'(x) * dx
Substituting the expression for dy into the expression for ds yields:
ds = √(dx^2 + (f'(x) * dx)^2)
Simplifying further, we get:
ds = √(1 + (f'(x))^2) * dx
Now, we can integrate ds to calculate the arc length on the curve. For any point x on the curve, we can express the arc length as:
L = ∫[x1,x2] √(1 + (f'(x))^2) dx (Just look at this formula)
By solving this integral, we can find the arc length L between two points A and B on the curve.
It should be noted that this method is only applicable to the case where the curve is continuously differentiable. If there are discontinuities or non-differentiable points in the curve, we need to consider other ways to calculate the arc length.
Method Two
In advanced mathematics, we can use derivatives to find the arc length of a curve. Specific steps are as follows:
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Suppose there is a curve, which can be expressed as a parametric equation : x = f(t), y = g(t), where t belongs to a certain interval.
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We first find the tangent vector to the curve: r'(t) = (f'(t), g'(t)).
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Calculate the modulus of the tangent vector: |r'(t)| = sqrt((f'(t))^2 + (g'(t))^2).
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Define the arc length function: s(t) = ∫[a, t] |r'(t)| dt , where a is the parameter value at a point on the curve.
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By deriving the arc length function s(t), the derivative of the arc length function of the curve can be obtained: ds/dt = |r'(t)|.
Through the above steps, we can use the derivative to find the derivative of the arc length function of the curve, so as to find the arc length of the curve.