# What is the net area between f(x) = 3x^2-x+2 and the x-axis over x in [1, 2 ]?

Apr 24, 2016

$7.5$ square units

#### Explanation:

First draw a rough graph of the function to visualize where the area is and to decide the limits of integration to find the area.

graph{3x^2-x+2 [-4.39, 9.66, -0.92, 6.107]}

Since $f \left(x\right) \ge 0 \forall x \in \mathbb{R}$, we may integrate directly over the entire given interval in order to find the area bounded, and it is given by

$A = {\int}_{1}^{2} \left(3 {x}^{2} - x + 2\right) \mathrm{dx}$

$= {\left[{x}^{3} - \frac{1}{2} {x}^{2} + 2 x\right]}_{1}^{2}$

$= \left(8 - 2 + 4\right) - \left(1 - \frac{1}{2} + 2\right)$

$= 7.5$