Chapter 15.5, Problem 13ES

Sometimes evaluating a surface integral results in an improper integral. When this happens, one can either attempt to determine the value of the integral using an appropriate limit or one can try another method. These exercises explore both approaches.

Consider the integral of $f\left(x,y,z\right)=z+1$ over the upper hemisphere $\sigma :z=\sqrt{1-x^2-y^2}\left(0\le x^2+y^2\le 1\right).$

(a) Explain why evaluating this surface integral using (8) results in an improper integral.

(b) Use (8) to evaluate the integral of f over the surface ${\sigma }_r:z=\sqrt{1-x^2-y^2}\left(0\le x^2+y^2\le r^2<1\right).$ Take the limit of this result as $r\to 1^{-}$ to determine the integral of f over $\sigma .$

(c) Parametrize $\sigma$ using spherical coordinates and evaluate the integral of f over $\sigma$ using (6). Verify that your answer agrees with the result in part (b).