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That can be a complicated process (although some curves are simpler than others). If the curvature can be described by an algebraic equation, then the area can be calculated by means of integral calculus. Alternatively, if you can fit simpler geometric shapes (such as squares and triangles) into the curved surface in a way that approximately fills that surface, you can use simpler means to calculate the area.

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Q: Finding area of a curved surface?
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