Applet: A parametrized helicoid with surface area elements
The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $(\spfv,\spsv) \in \dlr$, where $\dlr$ is the rectangle $[0,1] \times [0, 2\pi]$. The region $\dlr$, shown as the rectangle on the left, is divided into small rectangles, which are mapped onto “curvy rectangles” on the surface at the right. To visualize the mapping, you can click on a rectangle in either panel to highlight it in red; the corresponding rectangle in the opposite panel is also hightlighted in red. The area of the red region on the helicoid depends on how $\dlsp$ stretches or shrinks the small red rectangle of $\dlr$ as it maps it on the surface. You can also drag the blue point in either $\dlr$ or the helicoid surface, and the blue point in the opposite panel moves so that the location of the right point is $\dlsp(\spfv,\spsv)$ when the location of the left point is $(\spfv,\spsv)$.
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