**HP Prime and HP 17BII: Trapezoid Rule Using Distinct Points**

**Introduction**

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 ) )

Where:

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)

BEGIN

// h: increment between measurements

// ms: list of measurements

// 2019-05-09 EWS

LOCAL k,n:=SIZE(ms);

RETURN h/2 * (ms(1) + ms(n) + 2 * Σ( ms(k), k, 2, n-1 ));

END;

**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.

Solver:

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) )

Example

h = 0.5

MS:

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

Area: 3676

Eddie

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