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What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?

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They are when integrating the constant function 1.

Edit: The surface integral of the constant function 1 over a surface S equals the surface area of S. In other words, surface area is just a special case of surface integrals. A similar thing happens for line integrals: the line integral of the constant function 1 over a curve equals the length of the curve.

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Uh, which one? When you said "They" Do you mean <surface area of a parametrized surface> or <scalar surface integral of a function in $\mathbb{R}^3$> ?? – yiyi Dec 7 '12 at 1:24

The problem is when usually calculus teachers don't expand into multidimensional calculus in they way that might make sense. See a triple integral of a function might also seem strange until you realize that it is more like the volume integral of a 4 dimensional function. If you recall how a triple integral is turned into a double integral by integrating the function $f(x,y)$ over the region in question, in a same sense a 4 dimensional integral is turned into a triple integral over a three dimensional region (the function acts like the shadow of a 4 dimensional function in the same way as f acted like the shadow of a three dimensional function). This can be extended to surface integrals in the same sense, its simply the surface area of a higher dimensional object. A parameterized surface represents the normal three dimensional object whereas, a scalar function represents a higher dimensional object. And yes the above post is correct, when the function in question is constant at 1 they are the same.

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