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Put a string over the curved line. Measure the string.

If you want to do it by calculations for some known function, use integration. Divide the curved line into many small pieces, so that each piece can be approximated by a straight line, and use Pythagoras' Formula for each piece: length = square root of (width2 + height2), using the width and height of the piece. In calculus books, you'll find methods that can get you an exact result in some cases.

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Q: How can the length be measured along a curved line?
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