Question

Find the area of the region bounded by the graphs of y = x and
g(x) = x(1 – x^2) over the interval [0, 1].

Answer 1

the area underneath y=x on that interval is just 1/2 as its a triangle and for the other graph i would expand to give you g(x)=x-x^3 then you need to integrate over the interval to give you \[\int\limits_{0}^{1}x-x^3 dx=[x/2-x^4/4] \] from 0 to 1 which gives you 1/2-1/4=1/4. the area bounded by them is just there difference which is
1/2-1/4=1/4 u^2

Answer 2

or you could do \[\int\limits_{0}^{1}[x-x(1-x^2)]dx=\int\limits_{0}^{1}[x-x+x^3]dx=\int\limits_{0}^{1}x^3dx=\left.\frac{x^4}{4}\right|_{0}^{1}=\frac{1^4}{4}=\frac{0^4}{4}=\frac{1}{4}\]