if there are two regions enclosed by two lines how do I find area

Question

Equations

$y=x^3$ and $y=x$

obviously when graphing this, it would spit two areas formed.

Thus the "regular" way to set up the integral would be the following: $\int_{-1}^0(x^3-x)+\int_0^1(x-x^3)$ which gives me 0.5

But

how do I write this within one integral?

Answer 1

$$\int_{-1}^1 |x^3 - x|\text{d}x$$

Note that in order to actually solve this integral you'd still split the integral into the two pieces you originally had.

Answer 2

Absolute values. You only care about the distance between $x$ and $x^3$, so use the "distance between" operator (a.k.a. absolute value).

Answer 3

Alternatively, a change of variable (y=-x) yields $$ \int_{-1}^0 x^3-x\,dx =-\int_{1}^0 y-y^3\,dy =\int_{0}^1 y-y^3\,dy $$ So, $$ \int_{-1}^0(x^3-x)\,dx+\int_0^1(x-x^3)\,dx = 2\int_0^1(x-x^3)\,dx $$