Surface integral of a scalar function
We can parametrize a surface this way:
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Suppose that there is a function defined for all points on the surface. We wish to integrate over all points on surface . The surface integral of over the surface can be calculated
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In this interactive figure, move the smaller green point in the -region. You will see the value of at this point and you'll see the scaling factor . The integrand is the product of these two values.
Move the larger blue point from bottom left to top right to "integrate." When the blue point is in the top-right corner, the value of the surface integral is shown (this is a numerical approximation).
A technical note: Exact surface integral calculation can be quite difficult for most parametrized surfaces. In this applet we approximate the area using a midpoint Riemann sum. We lay out a 20 x 20 grid in the -region, and at each midpoint of the 400 subrectangles, the product times the scaling factor is observed. Each rectangle has area . We then add all the products together and multiply by . Because of this approximation, you'll notice that the integral calculation might not change with very small movements of the blue point.
This applet was developed for use with Interactive Calculus, published by Pearson.