Unfortunately, this is not true The accuracy of the trapezoidal rule is firmly related to the number of known curve points.

If the function is represented as a curve in a chart, then the integral is defined to be the (net signed) area under that curve.

One popular method for accomplishing this task is the so-called trapezoidal rule.

According to Wikipedia: The trapezoidal rule is a technique for approximating the definite integral: The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area.

It follows that: Calculate the area under a curvethe integral of a function 1st method: Spreadsheet calculations If n points (x, y) from the curve are known, you can apply the previous equation n-1 times and then sum the results.

In the sample workbook, for example, we had the function y 4x2, we knew 10 points, so we applied the formula 9 times.

For the first point the result was (1 0)(4 0)2 2, for the second (2 1)(16 4)2 10 and so on.

The picture above contains the entire set of calculations.

SUMPRODUCT formula With this method, you avoid the intermediate calculations, and by using only one function, you get the result.

However, the level of difficulty is a little bit higher than the first method (especially if you are new to Excel).

The method involves the SUMPRODUCT function, the syntax of which is given below: SUMPRODUCT(array1, array2, array3, ) The SUMPRODUCT function multiplies the corresponding components in the given arrays and returns the sum of these products.

Array1, array2 are the ranges of cells or arrays that you wish to multiply.

All arrays must have the same number of rows and columns, and you must enter at least 2 arrays (you can have up to 30 arrays).

If n curve points (x, y) are known, the function can be written: In the sample workbook the SUMPRODUCT function is used with the following ranges: SUMPRODUCT(A5:A13-A4:A12;(B5:B13B4:B12)2).

In reality, we applied the same function as in method 1, but instead of single cells, we had multiple cellsarrays.

The function performs the following calculation: Without any doubt, the second method is much more straightforward than the first one.

Custom VBA function At your Excel file, switch to VBA editor (ALT F11), go to the menu Insert Module and add the following lines of code.

Function CurveIntegration(KnownXs As Variant, KnownYs As Variant) As Variant.

End Function The advantage of this custom function is that you dont need to worry about the input ranges.

The code checks if the x and y values are (valid) ranges, as well as if the x and y ranges are equal (for example if the input is 10 x values and 10 y values).

If something goes wrong, the function returns an error message instead of the value.

So, you only have to insert the input ranges in the function, and the function will return the curve area.

In the sample workbook, for example, the custom function was used with the following ranges (we had 10 x values and 10 y values 10 curve points): CurveIntegration(A4:A13;B4:B13) Limitations of the trapezoidal rule As it was highlighted in the beginning, the trapezoidal rule is an approximate method to calculate the area under a curveto perform numerical integration.

In the sample workbook you will notice that, for the particular curve, all 3 different ways that were described above result in the same value (978).

You might believe that since the 3 methods agree on the final value, this is the correct one.

Integration Function In Excel