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?
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?
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?