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

1 Answer
Elijah T.
Feb 25, 2017

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⋅(12⋅b⋅h)
=2*(1/2*1*2)=2⋅(12⋅1⋅2)
=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 = 02x−6=0
x=3x=3 when y=0y=0

When x<3x<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=−∫322x−6dx+∫432x−6dx
=-[x^2-6x]_2^3 + [x^2-6x]_3^4=−[x2−6x]32+[x2−6x]43
=-[9-18-4+12] + [16-24-9+18]=−[9−18−4+12]+[16−24−9+18]
=-[-1] + [1]=−[−1]+[1]
=2=2 units squared. And Tada, that is the same as the other answer!

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