# How do you find the length of a curve using integration?

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To find the length of a curve using integration, follow these steps:

- Divide the curve into small segments, each represented by a straight line or a curve.
- Determine the length of each segment using the distance formula or arc length formula, depending on whether the segment is straight or curved.
- Sum up the lengths of all segments.
- Take the limit as the number of segments approaches infinity to get an integral.
- Set up the integral to represent the length of the curve.
- Integrate the appropriate expression over the interval of interest to find the length of the curve.

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To find the length of a curve using integration, you can follow these steps:

- Define the curve with a function ( y = f(x) ) or ( x = g(y) ), depending on whether it's described in terms of ( x ) or ( y ).
- Determine the interval over which you want to find the length of the curve.
- Use the formula for arc length, which is given by:

[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

or

[ L = \int_c^d \sqrt{1 + \left(\frac{dx}{dy}\right)^2} , dy ]

depending on whether the curve is described in terms of ( x ) or ( y ), respectively. 4. Integrate the expression within the square root over the given interval. 5. Evaluate the definite integral from the lower limit ( a ) (or ( c )) to the upper limit ( b ) (or ( d )).

This process yields the length of the curve over the specified interval.

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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.

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