Example 1: Definite Integral Power Rule

$\underset{3}{\overset{7}{\int }}6x+9\mathit{ dx}$

Step 1: Simplify and look for algebraic rewrites.

When looking at the first chunk Remind yourself that there is really an unwritten 1 for the power. That is what allows us to apply the power rule .

$\underset{3}{\overset{7}{\int }}6x^1+9\mathit{ dx}$

Step 2: Identify any term(s) that include variables raised to a power.

Break the problem down into bitesize chunks based upon the + and – , and identify the antiderivative rule for each chunk (term).

Here you have 2 chunks. The first chunk, 6x , will be a power rule , and the second chunk, 9, is a constant.

$\underset{3}{\overset{7}{\int }}6x^1+9\mathit{ dx}$

Step 3: Take the antiderivativeof the variables raised to a power using the Recipe:Add 1 to the power; Divide by the new power.

Chunk 1: Power Rule

Chunk 2: Constant Rule

$\underset{3}{\overset{7}{\int }}6x^1+9 \mathit{dx}\mathit{=}{\frac{6x^{1+1}}{2}+9x|}_{x=3}^{x=7}$

$\mathit{=}{\frac{6x^2}{2}+9x|}_{x=3}^{x=7}$

$\mathit{=}{3x^2+9x|}_{x=3}^{x=7}$

Step 4 ( Definite Integral ONLY ): Evaluate the antiderivative result using the Top – Bottom method.

$\underset{3}{\overset{7}{\int }}6x+9 \mathit{dx}\mathit{=}{3x^2+9x|}_{x=3}^{x=7}$

$\mathit{=}\left(3{\left(7\right)}^2+9\left(7\right)\right)\mathit{–}\left(3{\left(3\right)}^2+9\left(3\right)\right)$

$\mathit{=}\left(210\right)\mathit{–}\left(54\right)\mathit{=}\mathit{156}$

Final Result Meaning: Remember the Definite Integral will always provide you a definite value , and the Indefinite Integral provides you a family of solutions .

The Net Area between the curve $f\left(x\right)=6x+9$ and the x-axis on the x-interval [ 3 , 7 ] is 156.

Since the final result is positive, you know without even seeing the graph that there is more area above the x-axis than below it.