We can parametrize a surface [math]S[/math] this way:[br] [math]\mathbf{r}\left(u,v\right)=\left\langle f\left(u,v\right),g\left(u,v\right),h\left(u,v\right)\right\rangle[/math] [math]a\le u\le b[/math] [math]c\le v\le d[/math].[br]Suppose that there is a function [math]G\left(x,y,z\right)[/math] defined for all points [math]\left(x,y,z\right)[/math] on the surface. We wish to integrate [math]G[/math] over all points on surface [math]S[/math]. [b]The surface integral of[/b] [math]G[/math] [b]over the surface[/b] [math]S[/math] can be calculated[br] [math]\int_c^d\int_a^bG\left(f\left(u,v\right),g\left(u,v\right),h\left(u,v\right)\right)\left|\mathbf{r}_u\times\mathbf{r}_v\right|dudv[/math].[br][br]In this interactive figure, move the smaller green point in the [math]uv[/math]-region. You will see the value of [math]G[/math] at this point and you'll see the scaling factor [math]\left|\mathbf{r}_u\times\mathbf{r}_v\right|[/math]. The integrand is the product of these two values. [br][br]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).

[i]A technical note[/i]: 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 [math]uv[/math]-region, and at each midpoint [math]\left(u_0,v_0\right)[/math] of the 400 subrectangles, the product [math]G\left(f\left(u,v\right),g\left(u,v\right),h\left(u,v\right)\right)[/math] times the scaling factor [math]\left|\mathbf{r}_u\times\mathbf{r}_v\right|[/math] is observed. Each rectangle has area [math]\Delta u\Delta v=\frac{b-a}{20}\cdot\frac{d-c}{20}[/math]. We then add all the products together and multiply by [math]\Delta u\Delta v[/math]. Because of this approximation, you'll notice that the integral calculation might not change with very small movements of the blue point.[br][br][i]This applet was developed for use with [url=https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable/P200000009666]Interactive Calculus[/url], published by Pearson.[/i]