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Consider the integral of f(x, y, z) = z + 1 over the upper hemisphere σ:...

Consider the integral of f(x, y, z) = z + 1 over the upper hemisphere σ: z = 1 − x2 − y2 (0 ≤ x2 + y2 ≤ 1). (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 σr : z = 1 − x2 − y2 (0 ≤ x2 + y2 ≤ r2 < 1). Take the limit of this result as r →1− to determine the integral of f over σ. (c) Parametrize σ using spherical coordinates and evaluate the integral of f over σ using (6). Verify that your answer agrees with the result in part (b).

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