Exercise · ZW-SURFACE-INTEGRAL-010 · ★★★ Advanced
Let \(S\) be the cylinder \(x^2+y^2=1\), \(0\le z\le3\). The value of \(\iint_S z^2\,dS\) is:
Choose one option.
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- C. \(18\pi\)
Explanation
With \(z=v\) and \(dS=du\,dv\): \(2\pi\int_0^3v^2\,dv=2\pi\cdot9=18\pi\).
Review the concept: Surface integral
The surface integral of a function \(f\) over a surface \(S\), parametrized by \(\mathbf r\) over a region \(D\), adds up the values of \(f\) weighted by the area of each piece of \(S\).
\[\iint_S f\,dS=\iint_D f(\mathbf r(u,v))\,\lVert\mathbf r_u\times\mathbf r_v\rVert\,du\,dv\]
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