Infinite Area?

Calculus Level 2

For x ≥ 0 x\ge0 , find the area between two functions f ( x ) = e − x , g ( x ) = e − 2 x . \large f(x)=e^{-x}\quad,\quad g(x)=e^{-2x}.

The integral does not converge. -0.5 0.5 Infinite area. 1 0

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Leonardo de Araujo by Leonardo de Araujo , 32, Brazil

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1 solution

∫ 0 ∞ ( e − x − e − 2 x ) d x = [ − e − x + 1 2 e − 2 x ] 0 ∞ = ( − 0 + 0 ) − ( − 1 + 1 2 ) = 1 2 . \int_0^\infty (e^{-x}-e^{-2x})\:dx = \left[-e^{-x}+\frac12e^{-2x}\right]_0^\infty = (-0 + 0) - (-1 +\tfrac12) = \boxed{\tfrac12}.

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You should have specified that the area is to be taken only for the first quadrant. Otherwise the area will come to be infinite.

Prathmesh Pathwar - 5 years, 8 months ago

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The problem states x ≥ 0 x \geq 0 , so that limits the area to the first quadrant.

Arjen Vreugdenhil - 5 years, 8 months ago

It is specified on the problem (x>=0). And area can't be infinite, in this case it means that the integral (sum) diverges, so the limit doesn't exist.

Leonardo de Araujo - 5 years, 7 months ago

Perfect! But usually we don't let 0.5 and -0.5 to be different options of a question when areas are concerned although we usually have negative areas in integration.

Lu Chee Ket - 5 years, 7 months ago

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