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

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Answer 1 · 2017-12-15T23:16:07

See below.

Plugging in #x=2# and #x=4#

#2-4=-2# . #4-4=0#

shows us that the area we seek is under the x axis, and so will be negative.

We need to integrate #x-4#, with upper and lower bounds of #4# and #2#.

#A=-int_(2)^(4)(x-4) dx=[1/2x^2-4x]_(2)^(4)#

#A=-{[1/2x^2-4x]^(4)-[1/2x^2-4x]_(2)}#

Plugging in upper and lower bounds:

#A=-{[1/2(4)^2-4(4)]^(4)-[1/2(2)^2-4(2)]_(2)}#

#A=-{[-8]-[-6]}=2#

#Area = 2# square units

Graph:

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