Finding the area between curves is a common calculus problem, and one which demonstrates an actual use for the formulas you are studying. This area-between-two-curves problem is an incremental step on the learning path that had you first computing the area under a single curve between two points. In this problem, we compute the area under both of the provided curve functions and subtract the smaller area from the larger area to get our result. The general formula for this is the integral of u(x) - l(x) dx between given points a,b (where u is the upper curve and l is the lower curve).
Instructions
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1
Assume our upper function is a line y = x + 4 and our lower is a parabola y = x^2. We will find the area between these two functions between the x coordinates 0,2.
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2
Create our new formula by subtracting the upper function from the lower: Area = integral of (x + 4 - x^2) dx. This becomes -(1/3)x^3 + (1/2)x^2 + 4x
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3
Evaluate the area by substituting 0 and 2 into the formula and subtracting the values. Using 0 we get 0. Using 2 we get (-8/3) + (4/2) + 8 = 22/3.
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The solution is 22/3 - 0 = 22/3.