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Tutorial 13

Tutorial 13: Surface Integrals

Tutorial QuestionTutorial Solution (Early Access)

  1. Evaluate the surface integral of the vector field F=3x2i−2yxj+8k\mathbf{F}=3 x^{2} \mathbf{i}-2 y x \mathbf{j}+8 \mathbf{k} over the surface SS that is the graph of z=2x−yz=2 x-y over the rectangle [0,2]×[0,2][0,2] \times[0,2].

  2. Let SS be the triangle with vertices (1,0,0),(0,2,0)(1,0,0),(0,2,0) and (0,1,1)(0,1,1) and let F=xyz(i+j)\mathbf{F}=x y z(\mathbf{i}+\mathbf{j}). calculate the surface integral

    ∬SF⋅dS\iint_{S} F \cdot d \mathbf{S}

    If the triangle is oriented by the "downward" normal.

  3. The equations z=12,x2+y2≤25z=12, x^{2}+y^{2} \leq 25 describe a disk of radius 5 lying in the plane z=12z=12. Suppose that is the position vector field r(x,y,z)=xi+yj+zk\mathbf{r}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}. Compute ∬Sr.dS\iint_{S} \mathbf{r} . d \mathbf{S}.

  4. Let SS be the closed surface that consists of the hemisphere x2+y2+z2=1≥0x^{2}+y^{2}+z^{2}=1 \geq 0, and its base x2+y2≤1,z=0x^{2}+y^{2} \leq 1, z=0. Let EE be the electric field defined by E(x,y,z)=2xi+2yj+2zk\mathbf{E}(x, y, z)=2 x \mathbf{i}+2 y \mathbf{j}+2 z \mathbf{k}. Find the electric flux across SS.

  5. Find the area of the ellipse cut on the plane 2x+3y+6z=602 x+3 y+6 z=60 by the circular cylinder x2+y2=2xx^{2}+y^{2}=2 x.

  6. Find the integral ∬SxdS\iint_{S} x d S, where the surface SS is the part of the sphere x2+y2+z2=a2x^{2}+y^{2}+z^{2}=a^{2} lying in the first octant.

  7. Find the integral ∬SdSx2+y2+z2\iint_{S} \frac{d S}{\sqrt{x^{2}}+y^{2}+z^{2}}, where SS is the part of the cylindrical surface parameterized by r(u,v)=(acos⁡u,asin⁡u,v),0≤u≤2π,0≤v≤Hr(u, v)=(a \cos u, a \sin u, v), 0 \leq u \leq 2 \pi, 0 \leq v \leq H.