Down the Road to Infinity

Calculus Level 5

{ y = 0 y = ( 2 − x ) + x 2 − 4 y = ( 2 + x ) − x 2 + 4 \begin{cases} y=0 \\ y=(2-x)+\sqrt{x^2-4} \\ y=(2+x)-\sqrt{x^2+4} \end{cases}

Let the area bounded by the curves above be A A . Find ⌊ 100 A ⌋ \lfloor 100A\rfloor


The answer is 200.

Daniel Liu by Daniel Liu , 21, USA

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1 solution

Michael Mendrin
Aug 16, 2014

Look the problem the other way, solve for x x in both and get the following

x = y 2 − 4 y 2 y − 2 x=\dfrac { { y }^{ 2 }-4y }{ 2y-2 }

x = − y 2 + 4 y − 8 2 y − 2 x=\dfrac { -{ y }^{ 2 }+4y-8 }{ 2y-2 }

The difference is x = 2 − y x=2-y , which is just a straight line from ( 0 , 2 ) (0,2) to ( 2 , 0 ) (2,0) , and so the area of the triangle is 2 2 .

20 Helpful 0 Interesting 0 Brilliant 0 Confused

Yep, you got it!

Daniel Liu - 6 years, 10 months ago

I'm just using two paper full of calculus solving that, but you got a easier way!

Kelvin Hong - 3 years, 11 months ago

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