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“That book [Advanced Calculus, by Woods] also showed how to differentiate parameters under the integral sign – it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.”

What is this ‘differentiating under the integral sign’? Does anyone know this? Can anyone please help me?

And another thing Feynman wrote about is contour integration. What is this contour integration? Can anyone help me with that too?

In another chapter, Feynman wrote about computing the cube root of a number. The number was 1729.03. He wrote:

“I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03, is only one part in nearly 2000, and I have learned in calculus that for small fractions, the cube root’s excess is one-third of the number’s excess. So all I had to do is find the fraction 1/1728 and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.”

Can anyone explain that to me? What is the ‘cube root’s excess’? What is the ‘number’s excess’?

thanks in advance to anyone who can help.