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 $\dlr$ is shown as the green rectangle floating above the helicoid, and it is divided into small rectangles. As you drag the green point in $\dlr$ to specify both $\spfv$ and $\spsv$, a surrounding small rectangle is outlined in green. This rectangle is mapped into the small region of the helicoid that is outlined in red and surrounds the red point $\dlsp(\spfv,\spsv)$. The area of the red region on the helicoid depends on how $\dlsp$ stretches or shrinks the small green rectangle as it maps it on the surface.
Applet file: helicoid4.m
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