What is the net area between f(x) = 2x-6  and the x-axis over x in [2, 4 ]?

Feb 25, 2017

2 units squared.

Explanation:

Algebra

y = 2x-6 is linear, thus if we take the value over the x interval [2,4] we can use geometry to calculate the area.
graph{2x-6 [-2.27, 7.73, -2.18, 2.82]}

You can see two triangles in the graph (you could also find this algebraically). Thus, you can calculate the area.

$= 2 \cdot \left(\frac{1}{2} \cdot b \cdot h\right)$
$= 2 \cdot \left(\frac{1}{2} \cdot 1 \cdot 2\right)$
$= 2$ square units

Integrals

An integral gives the area under the curve. Remember though that if a function goes below the X axis, the integral is negative Thus you have to do two separate integrals based on the X intercept.

finding the X intercept (that is where y = 0)
$2 x - 6 = 0$
$x = 3$ when $y = 0$

When $x < 3$, then y is negative. Thus, we have to find the negative integral from 2 to 3 and the positive integral from 3 to 4

$= - {\int}_{2}^{3} 2 x - 6 \mathrm{dx} + {\int}_{3}^{4} 2 x - 6 \mathrm{dx}$
$= - {\left[{x}^{2} - 6 x\right]}_{2}^{3} + {\left[{x}^{2} - 6 x\right]}_{3}^{4}$
$= - \left[9 - 18 - 4 + 12\right] + \left[16 - 24 - 9 + 18\right]$
$= - \left[- 1\right] + \left[1\right]$
$= 2$ units squared. And Tada, that is the same as the other answer!