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Determine the area limited by curves:

$f(x)=2x^3-3x^2+9x \\ g(x)=x^3-2x^2-3x$

The correctly answer is: 25, How can I find it?

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    maybe he's being asked to give the anti-derivatives evaluated at fixed $a$ and $b$.2012-05-30

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Assuming that the limits of the interval are given $a, the fundamental theorem of calculus says that the "area" below the curve $f(x)$, and between $a$ and $b$ equals to $F(b)-F(a)$ where $F'(x)=f(x)$, $\forall x\in[a,b]$ (in case that such $F$ exists). Gladly, our $f$ & $g$ are polynomials and very easy to find an anti-derivative to (=indefinite integral).

So, we can find the anti-derivatives for $f(x)$ and $g(x)$ and evaluate the difference at $a$, $b$. If we let $S$ to be the area below $f(x)$ and above $g(x)$ we need to calculate the area below $f(x)$ and above the $X$ axis minus the area below $g(x)$ above the $X$ axis: $S = F(x)-G(x)|_{x=a}^{x=b}$

Do you know what's the anti-derivative of a Polynomial?

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    Notice that $\int f(x)+g(x) =\int f(x)+ \int g(x) $ so, you can integrate every "monic" by itself. Now, let $a\in\mathbb{R}$, $(a\cdot x^n)' = a\cdot n \cdot x^{n-1} \Rightarrow \int a\cdot x^{n-1} = a\frac{x^n}{n}$.2012-05-30