Determine the area between the curves f(x)=7x and g(x)=x^2-8

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We need to find out where they intersect, because those points will be our boundaries for integration.

Set them equal to one another:

7x = x^2 -8

x^2 - 7x - 8 = 0

Factor:

(x-8)(x+1) = 0

x = 8, x = -1

So! We need to subtract one function from the other (in this case, 7x is "on top" so to speak, in the first quadrant):

Integrate from -1 to 8 f(x) - g(x)

I(-1-->8) 7x - (x^2 - 8)

I(-1 --> 8) -x^2 + 7x + 8

I(-1 --> 8) -x^2/2 + 7x^2/2 + 8x

And you end up with 243/2 :)

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