What is the area between x^2 and x+2 ?

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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Let us assume that the functions are:

f(x) = x^2

g(x) = x+2

We need to find the area between both functions.

First we need to determine the intersection points.

==> f(x) = g(x)

==> x^2 = x+2

==> x^2 - x - 2 = 0

==> (x -2)(x+1) = 0

==> x= 2   x= -1

Then we need to find the area between x= -1 and x= 2

==> First we need to find the area under g(x).

==> G(x) = intg g(x) = intg (x+2) = x^2 /2 + 2x

==> A1 = G(2) - G(-1)

            = (4/2 +4) - ( 1/2-  2)

           = 6 +3/2 = 7.5

Now we need to find the area under f(x).

==> F(x) = intg f(x) = intg x^2 = x^3 /3

==> A2 = F(2) - F(-1)

            = 8/3 - -1/3 = 8/3 + 1/3 = 9/3 = 3

==> A = A1 - A2 = 7.5 - 3 = 4.5

Then, the are between x^2 and x+2 is 4.5 square units.

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