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Find the area between f(x)= 10000x^2 + x-1000 and g(x)= 2x-4 on [0, .322]. Don't round, and no graphing calculators or tools.

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First, we define our two given functions:

f(x) = 10000x^2 + x - 1000 and g(x) = 2x - 4

The area between these two functions over a set range can be found by taking the difference of these functions, taking the absolute value of that difference (to account for potential negative areas), and then integrating this over the given interval.

Let's denote the difference between the two functions as a new function h(x), so we have:

h(x) = f(x) - g(x)

Substituting the expressions for f(x) and g(x) into the equation for h(x) gives:

h(x) = (10000x^2 + x - 1000) - (2x - 4)

Simplify the equation to obtain a simplified form of h(x):

h(x) = 10000x^2 - x + 1004

Then we take the absolute value of h(x) (However, in this case, because on the interval [0, .322], h(x) is positive, so we ignore the absolute value function.), and integrate this over the interval from 0 to .322:

A = ∫ (from 0 to .322) h(x) dx

Performing this integration gives us the area between the two functions over the given interval, which in this case is approximately 209.476.

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