A Numerical Method for Computing Double Integrals with Variable Upper Limits
Round 1
Reviewer 1 Report
This work has been significantly improved over the previously submitted version. I agree with this publication and I wish the autors and the work all the best.
Author Response
Thank You very much.
Reviewer 2 Report
The authors have made all modifications so this paper gets the acceptance
Author Response
Thank You very much.
The position “English language and style are fine/minor spell check required” was marked.
We have made the necessary corrections to the article
Reviewer 3 Report
It is not necessary to use that many significant digits in your tables.
Author Response
We agree with the comment. Indeed, many significant digits complicate the analysis of numerical data. This presentation of the obtained results is caused by small errors of calculation, in particular, there are the errors of the order of 10-6 – 10-8. Accordingly, to show how the volumes of square and triangular elements change and their influence on the found numerical integrals, we need to take account of so many significant digits.
We propose to reduce, where possible, significant digits. The corresponding changes are made in tables 1, 3, 5, 6, 7, 8, 14, 15 and 16
The position “English language and style are fine/minor spell check required” was marked.
We have made the necessary corrections to the article.
Reviewer 4 Report
The main question adresed by the research is to approximate the double integral (1) which reduces to the integral (2). If $\tau$ is fixed then (2) is an integral on a triangle. We note that there exists literature for cubature on the triangle, even the remainder was studied. The authors also presented some numerical example. If we fix $tau$ in the double integral from the line 385 then we get a double integral on the rectangle. The cubature formulas on the rectangle were studied in some earlier papers. It is not very clear how the methods described in this paper improved the methods already existing in the literature.For the function from line 234 a cubature formula with the degree of exactness (1,1) gives better results.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
