Exercise · ZW-SURFACE-INTEGRAL-007 · ★★☆ Intermediate
If \(S\) is a flat region \(D\) in the \(xy\)-plane (parametrized by \(\mathbf r(x,y)=\langle x,y,0\rangle\)), then \(\iint_S f\,dS\) equals:
Choose one option.
A \(0\) B \(\sqrt2\iint_D f\,dA\) C \(\iint_D f(x,y,0)\,dA\) D \(\iint_D f\,dA\) plus the area of \(D\) Show answer C. \(\iint_D f(x,y,0)\,dA\)← Back to the question
Explanation Here \(\mathbf r_x\times\mathbf r_y=\langle0,0,1\rangle\) has length \(1\), so \(dS=dA\) and the surface integral is a double 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 \(S\).
\[\iint_S f\,dS=\iint_D f(\mathbf r(u,v))\,\lVert\mathbf r_u\times\mathbf r_v\rVert\,du\,dv\]
More practice problems: surface integral ZW-SURFACE-INTEGRAL-001 For mathbf r(u,v)=langle u, v, u+vrangle with 0≤ u,v≤1, the surface area is:ZW-SURFACE-INTEGRAL-002 For mathbf r(u,v)=langle u, v, u^2+v^2rangle, the area element dS is:ZW-SURFACE-INTEGRAL-003 The area of the cylinder x^2+y^2=1, 0≤ z≤3 (without caps) is:ZW-SURFACE-INTEGRAL-004 Let S be the cylinder x^2+y^2=1, 0≤ z≤3. The value of iint_S z dS is:ZW-SURFACE-INTEGRAL-005 Let S be the unit sphere. With dS=sin φ dφ dθ, the value of iint_S z^2 dS is:ZW-SURFACE-INTEGRAL-006 The area of a sphere of radius 3 is:ZW-SURFACE-INTEGRAL-008 The area of the part of the plane z=x+2y over the square 0≤ x,y≤1 is:ZW-SURFACE-INTEGRAL-009 The area of the part of the paraboloid z=x^2+y^2 with z≤1 is: