20181213

• I-Learn, Problem Set, Class Pictures

The two surfaces seen below are obtained by revolving the curve $z=y^2$ about the $z$-axis or the $y$-axis.

1. Consider the parametric surface $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
   • Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
   • Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
   • Set up an integral to compute the surface area of the surface.
2. Consider the parametric surface $\vec r(u,v) = (u^2\cos v, u, u^2\sin v)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
   • Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
   • Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
   • Set up an integral to compute the surface area of the surface.

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