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