Here is a routine which seems to do what you need.
An alternate approach woul have been to turn the table Temp/Signal into a function using some kind of interpolation (linear or spline). But to do so the vector Temp must consist of consecutive (sorted) values which is not the case in your sample data. So this problem would have to be addressed first by either sorting the vectors accordingly or cutting away the non sorted areas. The advantage with this approach would be that you could use the normal integral to get the area.
P.S.: Think you did a copy and paste to put the data values into this Prime table which seems not to be resizeable. I hide it in a collapsed region but wouldn't it be much more convenient (assuming you have more than one dataset to work on) to use an Excel component or simply READEXCEL() to get the data into the sheet?
EDIT: Appended second dataset via Excel component.