ยง16.5 Surfaces and Area
Math3d is recommended for experimenting with implicit surfaces!
Key equations
\[\begin{aligned} A &= \iint_R |\r_u\times\r_v|\,du\,dv\\[1em] A &= \iint_R \frac{|\nabla F|}{|\nabla F\cdot\mathbf p|}\,dA\\[1em] A &= \iint_R \sqrt{1+f_x^2+f_y^2}\,dx\,dy \end{aligned}\]Parametric vs implicit
Surface area
Surface area of implicit surfaces
Examples
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Find the surface area of a cone of height $h$ and base radius $r.$
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Find the surface area of a sphere of radius $r.$
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Find a parametrization of a torus of major radius $R$ and minor radius $r$. Then compute its surface area.
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Find the surface area of the band cut from the paraboloid \[ z = x^2 + y^2 \] by the planes $z=1$ and $z=4$.