What is the total area bounded by the curves y = x^5 and y = x^7 .?

Question

IS this correct 1/12 ???

Answer 1

Well, it depends on how you define the area.
(its either 1/12 or 0)

x^7 = x^5 means x = -1, 0 or 1
So there is an area between the curves in the interval [-1,0] and another in the interval [0,1]

∫ x^7 - x^5 dx represents the area difference between the curves.

x^8 / 8 - x^6 / 6 + C is the value of this integral.

If you just evaluate the integral in [-1,0] you get 1/24 , and if you evaluate it in [0,1] you get -1/24, and if you add them you get 0.
(The same as the integral from -1 to 1. This is true because the two regions are the same shape, but in [-1,0] x^5 < x^7,
and in [0,1] x^5 > x^7, so one is an "area", and the other is a "negative area". )

But if you are just looking for the geometric area of the two enclosed regions that are created by the two functions regardless of sign, then yes, they add to 1/12.

Steve, there are only two areas "bounded" by the curves.
(1, ∞) and (-∞,1) are unbounded regions.

Answer 2

You haven't specified any limits to the curves, so the answer is ∞.
The general expression would be

A = ⌠{(x^8)/8 - (x^6)/6}dx