Calculus/Calculus

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Expert: - 2/20/2010


Question
Hello Scotto. I had a question about solving for the area between curves. If f(x) = x^3-3x+3 and g(x) = 2x+1. I need to find the "total" area between them. How should I go about finding intersection points and integrating them? Thanks!

Answer
To find the area
1) set the equations equal
2) move all of elements to one side by subtracting off 2x+1 from both sides
3) solve for the intersection points
4) determine which function is higher
5) integrate the subtraction between the two points.

The differece gives x³ - 5x + 2 = 0.
This is divisible by x-2, and that gives (x-2)(x²+2x-1) = 0.
Factor the last one with a quadratic and get (-2±√(4+4)/2) = -1±√2.

Note that first the curvee is on top, and then the line is on top.  Points could be plugged in to determine the subtraction order.  An easier approach is to know since we are finding the area between the curve, it will be positive.  So if we get a negative result, the area would be minus that value (minus a minus is a plus).

Integrate the function x³ - 5x + 2 between -1-√2 and -1+√2, and find the absolute value of that.
Integrate the same function  from -1+√2 to 2 and find the absolute value of that.
Add the two positive quantities together.

This will be the total area.  

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Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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