=QUADXY(x, y, [options])
QUADXY
is a powerful function which employs splines to compute the integral of a function defined by a set of (x,y) data points.
With options, you can elect to weigh the data points, use exact or smooth least square fit, as well as specify end points slopes if known.
x
a vector of the points xcoordinates.
y
corresponding vector of the points yvalues.
ctrl
a set of key/value pairs for algorithmic control as detailed below.
Description of key/value pairs for algorithmic control
Key  ORDER 
Admissible Values (Integer)  1 or 3 (linear or cubic) 
Default Value  3 
Keys  ISLOPE, ESLOPE 
Admissible Values  real number 
Default Value  Unconstrained 
Remarks 

Key  PERIODIC 
Admissible Values (Boolean)  True or False 
Default Value  False 
Remarks 

Key  SFACTOR 
Admissible Values (real)  ≥ 0 
Default Value  0 
Remarks 

limits
integration lower and upper limits if different from the data set end points. Must be within data set. Passing a single value defines the lower limit only. passing a vector of 2 values defines both limits.
w
strictlypositive corresponding set of weights for the (x,y) data points. Default value is unity.
In this example we sample the function
$f\left(x\right)=\frac{2{x}^{5}x+3}{{x}^{2}}$, then
integrate the sampled data using QUADXY
and compare the result to the exact value:
$\underset{1}{\overset{2}{\int}}f\left(x\right)dx=9\mathrm{ln}2$
Define a vector for the x values in range A1:A21 from 1 to 2 in increments of 0.05. Using AutoFill,
generate the corresponding y values as shown in Table 1 from the formula =(2*A1^5 A1+3)/A1^2
in B1.
A  B  
1  1  4 
2  1.05  4.083957 
3  1.1  4.232248 
4  1.15  4.440616 
5  1.2  4.706 
6  1.25  5.02625 
7  1.3  5.399917 
8  1.35  5.8261 
9  1.4  6.304327 
10  1.45  6.834468 
11  1.5  7.416667 
12  1.55  8.051288 
13  1.6  8.738875 
14  1.65  9.480118 
15  1.7  10.27583 
16  1.75  11.12691 
17  1.8  12.03437 
18  1.85  12.99926 
19  1.9  14.02271 
20  1.95  15.10588 
21  2  16.25 
Table 2 shows the numerical integral value computed by QUADXY formula as well as the analytical exact integral value.
C  
1  =QUADXY(A1:A21,B1:B21) 
2  =9LN(2) 
C  
1  8.306853473 
2  8.306852819 
QUADXY
computes the integral by fitting a linear or cubic spline model to the data.