Evaluating a Definite Integral

Let's say you have the parabola ${\displaystyle x^{2}}$ and you want to find the area from x=2 to x=4

${\displaystyle 2\leq A\leq 4}$

${\displaystyle \int _{2}^{4}x^{2}\,dx}$

In order to take the integral of the function you have to do the opposite that of the derivative

The power of the variable x will have a number added to it. So, ${\displaystyle x^{(a+1)}}$

then the number gets inverted and brought out front.

${\displaystyle {\frac {1}{a+1}}*x^{(a+1)}}$

${\displaystyle \int _{2}^{4}x^{2}\,dx}$

From here we integrate and plug (b) into the indefinite integral and subtract the integral from (a) plugged into the indefinite integral.

${\displaystyle [{\frac {1}{3}}*4^{3}]-[{\frac {1}{3}}*2^{3}]}$

Now we evaluate the integral

${\displaystyle [{\frac {1}{3}}*64]-[{\frac {1}{3}}*8]}$

${\displaystyle [{\frac {64}{3}}]-[{\frac {8}{3}}]}$

${\displaystyle {\frac {56}{3}}}$

is the area underneath the curve from 2 to 4. In other words ${\displaystyle 2\leq A\leq 4}$