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December 29th, 2015, 12:20 AM   #1
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Integral calculation based on coordinates / Area under Curve calculation

Hello all,

I am looking for an easy solution to integral calculation based on coordinates. For example, what is the surface under: (0,5), (15,3), (30,2), (45, 5)?

Any suggestions on how to also automate the calculation in Excel of SPSS?

Thanks,
Joske
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December 29th, 2015, 05:02 AM   #2
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These are four points on a curve? We have to assume something about the graph between the points. The simplest thing to do is to assume a straight line between the points. Then finding the area under those lines is the "trapezoid method".

Dropping a line from each point perpendicular to the x-axis gives three trapezoids. The first has corners (0, 0), (0, 5), (15, 3), and (15, 0). That trapezoid has two parallel sides, from (0, 0) to (0, 5) and (15, 0) to (15, 3) of lengths 5 and 3, respectively. The "average base" is (5+ 3)/2= 4 while the perpendicular distance between bases, on the x axis, is 15. The area of that trapezoid is 4(15)= 60.

The next trapezoid has vertices at (15, 0), (15, 3), (30, 2), and (30, 0). Its two bases have length 3 and 2 so average 5/2. The distance between bases is again 15. The area of the second trapezoid is 15(5/2)= 75/2.

The last trapezoid has vertices at (30, 0), (30, 2), (45, 5), (45, 0). Its two bases have length 2 and 5 so average 7/2. The perpendicular distance between bases is 15. The area of the third trapezoid is 15(7/2)= 105/2.

The total area of the three trapezoids is 60+ 75/2+ 105/2= 150.
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