The actual numbers aren't completely relevant. I made a graphic.
I'm actually not in calculus or engineering. I'm actually writing a program, but I've done enough research to know how to differentiate and find areas under curves between x limits, just not between y limits as well.
I'm looking for the area of the green part of the graphic. Yes, I can and have solved it algebraically but I really need to be able to define this problem a certain way to apply to any situation, specifically:
I need to phrase this as an integral between two curves, within x limits and within y limits.
The specifics aren't completely necessary, but the x limits are x=1 & x=2. The slopes in this problem are 3x/4 and 384x/511. The y limits need to be y=384/641 to y=1152/1283.
The Attempt at a Solution
The areas under the curves are 3x^2/8 and 192x^2/511. The area under the first curve within x=0 and x=1 is 3/8, 1.5 between x=0 and x=2 and 1.125 between x=1 and x=2. The area under the second curve is 192/511, and the area between the two curves is 1149/1022, or 3063/4088 within x=1 and x=2.
I should add that I've set this up several ways as multiple integrals and not found the answer I was looking for.
There are any number of ways to solve this, obviously. But the situations could change. The slopes could be curves, and the needed x or y limits could change. That's why I need a specific way to define an area between two curves, within x limits and within y limits.
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