Numerical integration in one variable, of a multivariable interpolation function

Question

I create the following set of data:

newmatrix = Flatten[Table[{t, v, new[t, v]}, {t, 0, 2 Pi, Pi/4}, {v, 0, 1, 
0.01}], 1]

where

new[t_, v_] = t^2 + v^2

Then I defined a function that interpolate the previous data set:

newfun = Interpolation[newmatrix, Method -> "Hermite"]

I can plot it:

Now I need to integrate the interpolation function in one of the two variables like:

NIntegrate[newfun[t, v], {t, 0, 1}, AccuracyGoal -> 5]

but it doesen't work.

The error code is:

NIntegrate::inumr: The integrand <<1>> has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,0.785398}}.

Where is my error ?

Thanks for any tips and helps!

Answer 1

You can use Derivative to do your partial integration:

g = Derivative[-1, 0][newfun];

Visualization:

Plot[g[1, x], {x, 0, 1}]

Note that using Derivative in this way implies that the lower limit of integration is at the boundary of the interpolation domain.