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Section 7 Wrap Up Questions

  1. Are the following statements true or false? Give a brief reason.

    1. If the surface \(S\) is below the \(xy\)-plane, then the integral \(\int \int_{S} f(x,y,z)\,dS\) is negative.
    2. If \(S\) is the full sphere \(x^2+y^2+z^2=1\) and \(D\) the upper half-portion of that sphere, then \(\int \int_{S} f(x,y,z)\,dS=2\int \int_{D} f(x,y,z)\,dS\text{.}\)
    3. If \(f(x,y,z)\geq g(x,y,z)\) for all \((x,y,z)\text{,}\) then \(\int \int_{S} f(x,y,z)\,dS\geq \int \int_{S} g(x,y,z)\,dS\) for any surface \(S\text{.}\)
    4. The surface integral \(\int \int_{S} f(x,y,z)\,dS\) is independent of orientation of the surface \(S\text{.}\)
  2. Explain why even though a surface \(S\) lives in \(3\)-space \({\mathbb R}^3\text{,}\) the integral \(\int \int_{S} f(x,y,z)\,dS\) is a double integral and not a triple integral.

  3. Recall that a line integral of a scalar function \(f\) over a curve \(C\) parameterized by \(\vec{r}(t)\) for \(a\leq t\leq b\) is calculated using the formula

    \begin{equation*} \int_{C} fds=\int_{a}^{b} f(\vec{r}(t)) \, ||\vec{r}'(t)||\,dt. \end{equation*}

    Describe the differences and similarities to the surface integral \(\int \int_{S} f(x,y,z)\,dS\text{.}\) That is, describe how the formulas differ or share similarities and how the process of calculation differs or share similarities.

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