First, let's consider a rectangle. On a coordinate plane, we'll say it has vertices at (0,0), (a,0), (0,b), and (a,b)
Then, an orthogonal parametrization of the boundary can be obtained by taking F(x,y) = (ax, by)
The domain of F is D = {(x,y): 0 <= x <= 1, 0 <= y <= 1}
Our general formula claims that: Area = integral (sqrt [[ Fx*Fx, Fx*Fy][Fy*Fx, Fy*Fy]] over the domain D. The * operation in this case is the vector scalar product. Since the matrix is positive definite, the root of its determinant will never be imaginary so we will always have a positive or zero real area.
The partial derivatives can be easily computed to be
Fx = (a, 0) and Fy = (0, b)
Evaluating the determinant yields a^2*b^2, so that sqrt(a^2*b^2) = a*b
Integrating over [0,1] with respect to x and over [0,1] with respect to y yields Area = (a*b)*1*1 = a*b
So we see that the area of a rectangle is, perhaps not obviously, the product of the lengths of its sides.
[Edited on 12.10.2010 12:21 PM PST]
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Some one prove that V=(4/3)pi*r^3 for a sphere using Calculus! [Edited on 02.05.2011 10:59 AM PST]