The value of the surface integral S z\,dx\,dy where S is the external surface of the sphere x2 + y2 + z2 = R2 is

GATE 2024 ME – Question 12

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 1 mark · Multiple choice

The value of the surface integral

$$\oiint_S z\,dx\,dy$$

where $S$ is the external surface of the sphere $x^2 + y^2 + z^2 = R^2$ is

$0$
$4\pi R^3$
$\frac{4\pi}{3}R^3$
$\pi R^3$

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Correct answer: (C) $\frac{4\pi}{3}R^3$

Explanation

By the divergence theorem, $\oiint_S z\,dx\,dy = \iiint_V \frac{\partial z}{\partial z}\,dV = \iiint_V dV$, which is the volume of the sphere, $\frac{4\pi}{3}R^3$.