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

$2\leq A\leq 4$

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

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

then the number gets inverted and brought out front.

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

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

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

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

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 $2\leq A\leq 4$