Brilli the Ant

Calculus Level 3

Brilli the ant just found a Pringle on the ground. He decides it is a good place to start an ant colony. In order to do so, Brilli must compute the surface area of the Pringle. The Pringle has the shape of the surface z = x 2 − y 2 z=x^2-y^2 for x 2 + y 2 ≤ 1 x^2+y^2 \leq 1 . He finds that the surface area is

π ( a a − 1 ) b \frac{\pi (a \sqrt{a} - 1)}{b}

where a , b a, b are positive integers. Submit a + b a+b .


The answer is 11.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Tom Engelsman
Dec 3, 2017

The surface area in question is computed according to: ∫ − 1 1 ∫ − 1 − x 2 1 − x 2 1 + ( d z / d x ) 2 + ( d z / d y ) 2 d y d x ⇒ ∫ − 1 1 ∫ − 1 − x 2 1 − x 2 1 + 4 x 2 + 4 y 2 d y d x \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \sqrt{1 + (dz/dx)^2 + (dz/dy)^2} dy dx \Rightarrow \int_{-1}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \sqrt{1 + 4x^2 + 4y^2} dy dx which is much easier to calculate in cylindrical coordinates:

A = ∫ 0 2 π ∫ 0 1 r ⋅ 1 + 4 r 2 d r d θ = π ( 5 3 / 2 − 1 ) 6 . A = \int_{0}^{2\pi} \int_{0}^{1} r \cdot \sqrt{1 + 4r^2} dr d\theta = \frac{\pi(5^{3/2} - 1)}{6}.

Thus, a = 5 , b = 6 a = 5 , b = 6 and a + b = 11 . a + b = \boxed{11}.

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