Finding area between two bounded curves

**1. The problem statement, all variables and given/known data**

f(x) = (x^3) + (x^2) - (x)

g(x) = 20*sin(x^2)

**2. Relevant equations**

**3. The attempt at a solution**

I found the zeroes of the two functions at 4 intersections, and then the zeroes of each function respectively (there's 3 for f(x) and 4 for g(x) between -3 and 3), for certain reference points when I'm making the integration.

I did:

integral of [-g(x) from g(0)_1 to g(0)_2] - integral of [-g, x = g(0)_1 intersection_1] - integral of [-f from intersection_1 to intersection_2] - integral of [-f from intersection_2 to g(0)_2]

+ integral of [-f from intersection_2 f(0)_1] - integral of [-g from intersection_2 to g(0)_2]

+ integral of [g from g(0)_2 to 0] - integral of [f from f(0)_1 to 0]

+ integral of [-f from 0 to f(0)_3]

+ integral of [g from 0 to intersection_4] - integral of [f from f(0)_3 to intersection_4] - integral of [g from intersection_4 to g(0)_4]

I used a graph on the Wolframalpha website as a guide. There's 5 main parts to take the areas of and subtract the respective unnecessary areas. I got 39.19 as my answer but I have no way of checking my solution so I wanted to make sure with more knowledgeable people.

Re: Finding area between two bounded curves

My TI-89 gives about 39.4548, using the command:

∫(abs(x^3+x^2-x-20sin(x^2)),x,-2.44325,1.68523)