Exercise · ZW-SURFACE-INTEGRAL-005 · ★★☆ Intermediate
Let \(S\) be the unit sphere. With \(dS=\sin\varphi\,d\varphi\,d\theta\), the value of \(\iint_S z^2\,dS\) is:
Choose one option.
Show answer
- D. \(\dfrac{4\pi}{3}\)
Explanation
\(z=\cos\varphi\), so \(\int_0^{2\pi}\!\int_0^\pi\cos^2\varphi\sin\varphi\,d\varphi\,d\theta=2\pi\cdot\tfrac23=\tfrac{4\pi}{3}\).
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-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-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: