Question

How do you find the area under the graph of `f(x)=x^3` on the interval `[-1,1]` ?

Answer 1

Well, we need to be careful when you say "under the graph" since `f(x)=x^3` goes below the x-axis when `x<0`, but you meant the region between the graph and the x-axis, then the area of the region is 1/2.

Since there are two regions: one from `x=-1` to `x=0` and the other from `x=0` to `x=1`, the area `A` can be found by
`A=int_{-1}^0(0-x^3)dx+int_0^1(x^3-0)dx`
by Power Rule,
`=[-x^4/4]_{-1}^0+[x^4/4]_0^1=1/4+1/4=1/2`