MA Day 61 – April 12, 2013 Pages : Tangent planes to parametric surfaces – an example Section 12.6: Surface area of parametric surfaces – Review and examples Section 13.6: Surface integrals

Let S be the parametric surface traced out by the vector- valued function as u and v vary over the domain D. Pages : Tangent planes to parametric surfaces

x

x

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Section 12.6: Surface area of parametric surfaces

Goal: To compute the surface area of a parametric surface given by with u and v in domain D in the uv-plane. 1. Partition the region D, which also partitions the surface S

So we approximate by the Parallelogram determined by and

So we approximate by the Parallelogram determined by and

Now find the surface area.

Another method:

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Section 13.6: Surface Integrals

Section 12.6: Surface area of parametric surfaces Goal: To define the surface integral of a function f(x,y,z) over a parametric surface given by with u and v in domain D in the uv-plane.

Section 12.6: Surface area of parametric surfaces Goal: To define the surface integral of a function f(x,y,z) over a parametric surface given by with u and v in domain D in the uv-plane. 1. Partition the region D, which also partitions the surface S

Section 12.6: Surface area of parametric surfaces

How do we evaluate such an integral?

Recall our approximation of surface area:

The surface integral over S is the “double integral of the function over the domain D of the parameters u and v”.

This formula should be compared to the line integral formula

Notice the special case: The surface integral of f(x,y,z) = 1 over S yields the “surface area of S”

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