High School Calculus/Evaluating Definite Integrals

Evaluating a Definite Integral

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

$2\le A\le 4$

$\underset{2}{\overset{4}{\int }}{x}^{2}dx$

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, $x^{(a+1)}$

then the number gets inverted and brought out front.

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

$\underset{2}{\overset{4}{\int }}{x}^{2}dx$

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

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

Now we evaluate the integral

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

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

$\frac{56}{3}$

$\frac{56}{3}$

is the area underneath the curve from 2 to 4. In other words $2\le A\le 4$