Exercise · ZW-SURFACE-INTEGRAL-003 · ★☆☆ Basic
The area of the cylinder \(x^2+y^2=1\), \(0\le z\le3\) (without caps) is:
Choose one option.
Show answer
- D. \(6\pi\)
Explanation
With \(\mathbf r=\langle\cos u,\sin u,v\rangle\), \(\lVert\mathbf r_u\times\mathbf r_v\rVert=1\) and \(D=[0,2\pi]\times[0,3]\): the area is \(2\pi\cdot3=6\pi\).
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 \(S\).
\[\iint_S f\,dS=\iint_D f(\mathbf r(u,v))\,\lVert\mathbf r_u\times\mathbf r_v\rVert\,du\,dv\]
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-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-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: