Find the area of the region bounded by the graphs of y=x^2 and y=x^2+4x

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the limits of integration solving the following equation such that:

`x^2 = x^2 + 4x => 4x = 0 => x = 0`

Hence, since the supper limit cannot be determinated, you may consider it oo.

You should notice that `x^2 + 4x > x^2`  for `x in (0,oo), ` hence, you may evaluate the area of the region bounded by the given curves such that:

`int_0^oo (x^2 + 4x - x^2)dx = lim_(n->oo) int_0^n 4x dx`

`lim_(n->oo) int_0^n 4x dx = lim_(n->oo) (4x^2/2)|_0^n`

`lim_(n->oo) (2x^2)|_0^n = lim_(n->oo) 2n^2 = 2*oo = oo`

Hence, evaluating the area of the region bounded by the given graphs of functions, under the given conditions, yields `int_0^oo (x^2 + 4x - x^2)dx = oo.`

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