We can parametrize a surface 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]This surface is [b]smooth [/b]if the partial derivatives [math]\mathbf{r}_u[/math] and [math]\mathbf{r}_v[/math] are continuous and if [math]\mathbf{r}_u\times\mathbf{r}_v[/math] is never zero in the interior of the parameter domain [math]R[/math]. If the parametrization defines a smooth surface, then the area of the surface is [br] [math]A=\int\int_R\left|\mathbf{r}_u\times\mathbf{r}_v\right|dA=\int_c^d\int_a^b\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 to see the scaling factor [math]\left|\mathbf{r}_u\times\mathbf{r}_v\right|[/math] at each point. The vector [math]\mathbf{r}_u\times\mathbf{r}_v[/math] can also be shown. It is perpendicular to the surface, and it's nice to consider that the length of that vector is the scaling factor of the parametrization at that point.[br][br]Move the larger blue point from bottom left to top right to accumulate area. When the blue point is in the top-right corner, the area of the entire surface is shown.

[i]A technical note[/i]: Exact area 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 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 scaling factors and multiply by [math]\Delta u\Delta v[/math]. Because of this approximation, you'll notice that the area calculation might not change with very small movements of the blue point.

[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]