Problem 1371 (difficulty: 7/10)

Let \(\displaystyle f:\R^3\to\R\) be continuous, and \(\displaystyle G(r)=\int_{x^2+y^2\le r^2}f(x,y,r)\dx\dy\) (\(\displaystyle r>0\)).

(a) Show that \(\displaystyle G\) is continuous.

(b1) Show that if \(\displaystyle f\) continuously differentiable, then \(\displaystyle G\) is also continuously differentiable. What is \(\displaystyle G'\)?

(b2) Can the condition of continuous differentiablity be weakened?

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