The infinite plot with a finite area

Algebra Level 4

Let x x be the sum of the areas formed when plotting the given equations:

{ a 2 + b 2 ≥ ( a + b ) 2 a 2 + b 2 ≤ 1. \begin{cases} a^2 + b^2 \geq (a + b)^2\\ a^2 + b^2 \leq 1. \end{cases}

Find the value of ( e i x ) 2 . (e^{ix})^2.

Details and Assumptions

  • e e is the base of the natural logarithm.

  • i i is the imaginary unit i = − 1 . i=\sqrt {-1}.


The answer is -1.

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Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Eddie The Head
Apr 23, 2014

The first equation corresponds to the entire 2nd and 4th quadrant and the second equation corresponds to a circle centered at origin .So area of intersection is = x = π / 2 x = \pi/2 .

Putting this value in ( e i x ) 2 (e^{ix})^{2} we get e i π = − 1 e^{i\pi} = -1 .

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