We can estimate the area of any surface by the use of sums and integral. In calculus, we usually are given a function f(x), but here we are using measurements from one end to the other at various intervals.
Technically, the intervals between each measurement do not have to be equal length. However, having intervals of equal length makes things a lot easier, and in this blog entry, we assume they are.
We have various methods to estimate the area. One of the easiest ways is the Trapezoid Rule:
Area ≈ h/2 * ( y_1 + y_n + 2 * Σ( y_k , k, 2, n-1 ) )
h = interval length
y_k = length of each measurement, there are n measurements
y_1 and y_n: measurement of lengths at each end, respectively
Another rule to estimate area is the Simpson’s Rule:
Area ≈ h/3 ( y_0 + y_n + 4 * Σ( y_k, k, 1, n-1, 2) + 2 * Σ( y_k, k, 2, n-2, 2) )
The program presented here uses the Trapezoid Rule.
HP Prime Program AREAHGT
Two arguments: h, a list of measurements
EXPORT AREAHGT(h, ms)
// h: increment between measurements
// ms: list of measurements
// 2019-05-09 EWS
RETURN h/2 * (ms(1) + ms(n) + 2 * Σ( ms(k), k, 2, n-1 ));
HP 17BII+ (Silver)/HP 17BII Solver: Trapezoid Rule
First: define a SUM list named MS. The solver uses that list to get the reference measurements.
AREAHGT: AREA = 0 * L(N:SIZES(MS)) + H÷2 * (ITEM(MS:1) + ITEM(MS:G(N)) + 2 * Σ(K:2: G(N)-1: 1: ITEM(MS:K) )
h = 0.5
y_1 = 1174
y_2 = 1078
y_3 = 979
y_4 = 984
y_5 = 810
y_6 = 779
y_7 = 800
y_8 = 852
y_9 = 966
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