 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/4/b/solution

The relevant projection is the disk

x^{2} + y^{2} \leq 9

\iint 1 + z_{x}^{2} + z_{y}^{2} d x d y

. Here

z_{x} = - 2 x

and

z_{y} = - 2 y

A = 2 π \int_{0}^{3} r 1 + 4 r^{2} d r = \frac{π}{6} [(1 + 4 r^{2})^{3/2}]_{0}^{3} = \frac{π}{6} (3737 - 1) .

The boundary circle has zero surface area, so including the points at height zero does not change the result.

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