High School Calculus/Evaluating Definite Integrals

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Evaluating a Definite Integral[edit]

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