Area of region under curves?

Question

Find area of region enclosed by the curves.
Two curves are x-y^(2/3)=0 and x+y^4=2.

I know that in this case we should set y^(2/3) = 2-y^4. The problem is, how do I solve for y? Two solutions are +1, -1 but how do I continue from here w/o using a calculator.

Answer 1

If you are looking for the area of the region between the two curves, you have to integrate.

Let F(u) and G(u) be two functions of u with:

F(a) = G(a)
F(b) = G(b)
F(u) <= G(u) for all u in [a,b]
F(u) > G(u) for all other u

Then in most cases (including yours), the area of interest (i.e. between the curves) would be:

Area = the integral from a to b of G(u)-F(u) du

So first you have to decide whether you want y as a function of x or vice versa; then you have to determine a and b, and finally, you have to compute the definite integral.