How do you find the length of a curve in calculus?
In Cartesian coordinates for y = f(x) defined on interval
In general, we could just write:
Let's use Cartesian coordinates for this explanation.
This means that the approximate total length of curve is simply a sum of all of these line segments:
which we could also write (using the notation we are using) as
Applying this means we now have
Simplifying this expression a bit gives us
We can now use this new distance definition for our points in our summation.
Sums are nice, but integrals are nicer for continuous circumstances! It's easy to just write this as a definite integral since both integrals and sums are "summation" tools. In the integral, we can drop our sum index as well.
Writing this a little bit more typically yields
In general, you need to take the derivative of the function defining your curve to substitute into the integral. Then the trick is to find a way (usually) to try and get a perfect square inside the square root to simplify the integral and find your solution. It varies for every type of curve.
Let me know if you have any further questions in the comments!
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To find the length of a curve in calculus, you use the arc length formula. For a curve defined by the function y = f(x) from x = a to x = b, the arc length (L) is given by the integral:
L = ∫[a, b] √(1 + (f'(x))^2) dx.
Where f'(x) represents the derivative of the function with respect to x. This formula calculates the distance along the curve from the starting point (x = a) to the ending point (x = b).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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