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Exercises 3.6 Exercises

1.

Compute
\begin{equation*} \iint_R (2x-y)\,dA, \qquad R=[0,2]\times[-1,1]. \end{equation*}

2.

Let
\begin{equation*} D=\{(x,y): 0 \le x \le 2,\ 0 \le y \le x\}. \end{equation*}
  1. Write \(\iint_D (x+y)\,dA\) as an iterated integral with the order \(dy\,dx\text{.}\)
  2. Rewrite the same integral with the order \(dx\,dy\text{.}\)
  3. Evaluate the integral.

3.

Compute
\begin{equation*} \iiint_B xyz\,dV, \qquad B=[0,1]\times[0,2]\times[0,3]. \end{equation*}

4.

Use polar coordinates to compute
\begin{equation*} \iint_D \sqrt{x^2+y^2}\,dA, \end{equation*}
where \(D=\{(x,y): x^2+y^2 \le 1\}\text{.}\)

5.

Use cylindrical coordinates to compute the volume of the solid cone
\begin{equation*} E=\{(x,y,z): x^2+y^2 \le z^2,\ 0 \le z \le 2\}. \end{equation*}

6.

Use spherical coordinates to compute
\begin{equation*} \iiint_{B_a} (x^2+y^2+z^2)\,dV, \end{equation*}
where \(B_a=\{(x,y,z): x^2+y^2+z^2 \le a^2\}\text{.}\)

7.

Let
\begin{equation*} \mathbf{F}(x,y,z)=\langle xz,\; yz,\; xy \rangle. \end{equation*}
Compute \(\nabla \cdot \mathbf{F}\) and \(\nabla \times \mathbf{F}\text{.}\)

8.

Let
\begin{equation*} \omega = (2xy+\cos y)\,dx + (x^2-x\sin y)\,dy \end{equation*}
on \(\R^2\text{.}\)
  1. Compute \(d\omega\text{.}\)
  2. Show that \(\omega\) is exact.
  3. Find a potential function \(f\) such that \(df=\omega\text{.}\)

9.

Let \(\Phi(r,\theta)=\langle r\cos\theta,\; r\sin\theta \rangle\) be the polar-coordinate map, and let
\begin{equation*} \omega=x\,dx+y\,dy. \end{equation*}
Compute \(\Phi^*\omega\text{.}\)

10.

Let
\begin{equation*} c(t)=\langle t,\; t^2 \rangle, \qquad 0 \le t \le 1, \end{equation*}
and let
\begin{equation*} \omega=(x+y)\,dx+x\,dy. \end{equation*}
Compute \(\int_c \omega\text{.}\)

11.

Let \(\mathbf{F}(x,y)=\langle 2x+y,\; x+2y \rangle\text{.}\)
  1. Show that \(\mathbf{F}\) is conservative on \(\R^2\text{.}\)
  2. Find a potential function for \(\mathbf{F}\text{.}\)
  3. Compute \(\int_{\gamma}\mathbf{F}\cdot d\mathbf{r}\text{,}\) where \(\gamma\) is any piecewise \(C^1\) curve from \(\langle 0,0 \rangle\) to \(\langle 1,2 \rangle\text{.}\)

12.

Let
\begin{equation*} \sigma(u,v)=\langle u,\; v,\; u+v \rangle, \qquad 0 \le u,v \le 1. \end{equation*}
  1. Compute the surface area of \(\sigma\text{.}\)
  2. Compute the upward flux of \(\mathbf{F}=\langle 0,0,1 \rangle\) across \(\sigma\text{.}\)

13.

Explain how general Stokes’ theorem with \(p=1\) reduces to the one-variable Fundamental Theorem of Calculus.

14.

Let \(C\) be the boundary of the unit square \([0,1]\times[0,1]\text{,}\) oriented counterclockwise. Use Green’s theorem to compute
\begin{equation*} \oint_C (x^2-y)\,dx + (x+y^2)\,dy. \end{equation*}

15.

Let \(C\) be the boundary of the disk \(x^2+y^2 \le 1\) in the plane \(z=0\text{,}\) oriented counterclockwise as viewed from above. Use Stokes’ theorem to compute
\begin{equation*} \oint_C \mathbf{F}\cdot d\mathbf{r}, \qquad \mathbf{F}(x,y,z)=\langle -y,\; x,\; z \rangle. \end{equation*}

16.

Use the divergence theorem to compute the outward flux of
\begin{equation*} \mathbf{F}(x,y,z)=\langle x^3,\; y^3,\; z^3 \rangle \end{equation*}
across the boundary of the cube \([0,1]^3\text{.}\)
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