Exercise · ZW-SURFACE-INTEGRAL-008 · ★★☆ Intermediate
The area of the part of the plane \(z=x+2y\) over the square \(0\le x,y\le1\) is:
Choose one option.
Show answer
- B. \(\sqrt6\)
Explanation
For \(z=f(x,y)\), \(dS=\sqrt{1+f_x^2+f_y^2}\,dA=\sqrt{1+1+4}\,dA=\sqrt6\,dA\), and the square has area \(1\).
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 \(S\).
\[\iint_S f\,dS=\iint_D f(\mathbf r(u,v))\,\lVert\mathbf r_u\times\mathbf r_v\rVert\,du\,dv\]
More practice problems: surface integral
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- 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-007 If S is a flat region D in the xy-plane (parametrized by mathbf r(x,y)=langle x,y,0rangle), then ii…
- ZW-SURFACE-INTEGRAL-009 The area of the part of the paraboloid z=x^2+y^2 with z≤1 is: