Question

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

Answer 1

2 units squared.

Explanation:

You can go about this in two ways: Algebra and Integrals.

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*(1/2*b*h)`
`=2*(1/2*1*2)`
`=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)
`2x-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 2x-6 dx + int_3^4 2x-6 dx`
`=-[x^2-6x]_2^3 + [x^2-6x]_3^4`
`=-[9-18-4+12] + [16-24-9+18]`
`=-[-1] + [1]`
`=2` units squared. And Tada, that is the same as the other answer!