Finding area between two bounded curves

PhizKid

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.

haruspex

It's not clear to me that the zeroes of the functions have anything to do with it. The title says the area between the curves, not between the curves and some lines parallel to the axes. OTOH, my reading of it implies you need to solve f=g to find the range of integration, and I don't believe there's an analytic solution to that. Further, I've no idea how to integrate sin(x²).
Are you supposed to do this analytically or by approximation?

LCKurtz

You haven't stated what the question is that you are working on.

What do these strange notations like g(0)_1 mean?

See above.

[Edit -- added later] If the problem is to find the finite area that is bounded by the curves I get 39.45480254 using Maple.

haruspex

Given the verbal description, I assumed they meant the various roots of g(x)=0, g(0)_1 being the 'leftmost'.

HallsofIvy

Since there are, in fact, an infinite number of intersections of the two curves, I suspect there was some restriction on x that we were not told.

PhizKid

I think you need the zeroes to find certain areas; for example the area in between the curves from ~-1.782343436 to -1.618033989. -1.618033989 is a zero of the f(x) function, otherwise I don't know how else to break up the parts without the zeroes.

I have to find the area between the bounded curves of the two functions

The first zero of the g(x) function that appears right before the first intersection of the two curves. I should have been more clear on that but around the interval [-3,3] would be good.

I am getting 39.45388299 in Maple by doing each integration piece by piece. I am not allowed to perform something like: int(abs((x^3+x^2-x) - (20*sin(x^2))), x = intersection1 .. intersection4);

Which piece(s) and I calculating incorrectly?

Yes

I only calculated 4 intersections. How come there are an infinite number of them?

haruspex

The ranges are defined by the intersections of the curves, i.e. the roots of f-g. They have nothing to do with the roots of f and g individually.
You're interpreting the question as ∫|f-g|dx, which is probably right, so you do need to find all intersections and integrate separately between each successive pair. But in a mathematical sense areas can be negative, so it could reasonably be interpreted as |∫(f-g)dx|.
I count 5. There must be a finite odd number since x³ is asymptotically +∞ one side and -∞ the other, while the 20*sine function is bounded. Two are very close together. There's 0, obviously, and another at -.05.

PhizKid

Wow, this was exactly my problem. I missed that -0.05 intersection!!
Thank you