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\noindent {\bf 21-259} \hfill {\bf Calculus in 3-D} \hfill {\bf Spring 2003}

\centerline{\bf Review Problems for Test 3} 



\begin{enumerate} 

\item Evaluate each of the following. 

\begin{enumerate} 

\item $ {\displaystyle \int^1_0}\ {\displaystyle \int^y_0} \left({x^2+y^2}\right)\ dxdy$ 

\item ${\displaystyle \int^2_{-2}} \ { \displaystyle \int^{\sqrt{4-x^2}}_0} e^{x^2+y^2}\ dy dx$

\end{enumerate} 

\item Let $D$ be the region in the first quadrant that is bounded by $y = \sqrt{1-x^2},\ y=x$, and the $x$-axis.  Evaluate 

$${\displaystyle \int_D} {\displaystyle \int} (1+\sqrt{x^2+y^2})\ dA$$ 

\item Find the volume of the region enclosed by $y=x^2,\ y=1,\ z=0$,\ and $z=2$. 

\item Find the volume of the region that is inside the sphere $x^2+y^2+z^2=4$, but outside the cylinder $x^2+y^2=1$. 



\item Let $E$ be the region in the first octant that is bounded by the coordinate planes, the plane $x+y=1$ and the plane $z=3$.  Evaluate 

$${\displaystyle \int} {\displaystyle \int_E}{\displaystyle \int} (1+2z)dV.$$ 

\item Let $E$ be the region in the first octant that is bounded by the coordinate planes and the sphere $x^2+y^2+z^2=1$.  Evaluate 

$${\displaystyle \int} {\displaystyle \int_E} {\displaystyle \int} (x^2+y^2+z^2)^{3/2}dV.$$ 



\item Evaluate each of the following line integrals. 

\begin{enumerate} 

\item ${\displaystyle \int_C} \vec{F} \cdot d \vec{r}$, where $\vec{F} (x,y,z)=y\ \vec{i}-x\ \vec{j}+(x^2+y^2+z^2)\vec{k}$ and $C$ is described by  $\vec{r}(t) = \sin \vec{i} + \cos t\ \vec{j} + t\ \vec{k},0 \leq t \leq \pi$. 

\item ${\displaystyle \int_C} \left(\vec{\nabla}f\right) \cdot d \vec{r}$, where $f(x,y,z)=xyz$ and $C$ is described by\\ $\vec{r} (t) = (1+t^4) \vec{i} + t^3\ \vec{j}+e^t\ \vec{k}, 0 \leq t \leq 1$. 

\item ${\displaystyle \int_C} \vec{F} \cdot d \vec{r}$, where $\vec{F} (x,y,z) = (yz+x)\vec{i} + xz\ \vec{j} + xy \ \vec{k}$, and $C$ is described by $\vec{r} (t) t = t\ \vec{i} +t^2\ \vec{j} + t\ \vec{k}, 0 \leq t \leq 1$. 

\end{enumerate} 

\item Evaluate each of the following line integrals. 

\begin{enumerate} 

\item ${\displaystyle \int_C}ydx+xydy$, where $C$ is the segment of $y=x^3$ from $(0,0)$ to $(1,1)$. 

\item ${\displaystyle \int_C}-ydx+(x+y^2)dy$, where $C$ is the segment of $x^2+y^2 =1$ from (1,0) to (0,1). 

\end{enumerate} 

\item Find a function $f(x,y)$ such that $\vec{\nabla} f(x,y) =(2xy+x) \vec{i} + (x^2+y^3) \vec{j}$. 

\item Evaluate 

$${\displaystyle \int_C}(e^y+xy)dx + xe^ydy, $$ 

where $C$ is the segment of $y=x^2$ from (0,0) to (1,1).  

\item Let $C$ be a piecewise smooth simple closed curve in the $x-y$ plane and let $D$ be the region enclosed by $C$.  Given that 

$${\displaystyle \int_D} {\displaystyle \int} dA = \pi, {\displaystyle \int_D}{\displaystyle \int} ydA=3,\ {\displaystyle \int_C}x^2dy=2,$$

evaluate 

$${\displaystyle \int_C}(xy+2y+1)dx+(y^2+3xy+5x)dy.$$ 

\item Use Green's Theorem to evaluate ${\displaystyle \int_C}y^3dx - x^3dy$ where $C$ is the circle \newline $x^2+y^2=1$.  

{\bf In Problems 11-13, $C$ is oriented counterclockwise.}

\item Let $C$ be a piecewise smooth simple closed curve in the $x-y$ plane and let $D$ be the region enclosed by $C$.  Show that if $u(x,y)$ is a smooth function satisfying $u_{xx}+u_{yy}=0$ in $D$ then 

$${\displaystyle \int_D}{\displaystyle \int} \left(u^2_x + u^2_y\right)dA = {\displaystyle \int_C}- uu_y dx+uu_x dy. $$

\item Let $S$ be the portion of the cone $z=4-\sqrt{x^2-y^2}$ for which $1 \leq x^2+y^2 \leq 4$. 

\begin{enumerate} 

\item Find the surface area of $S$. 

\item Evaluate ${\displaystyle \int_S} {\displaystyle \int} \left(z+2 \sqrt{x^2+y^2}\right)dS$

\end{enumerate} 

\item Let $S$ be the hemisphere $z=\sqrt{9-x^2-y^2}$ and let $\vec{n}$ be the unit normal to $S$ that points upward.  Evaluate 

$${\displaystyle \int_S} {\displaystyle \int} \vec{F} \cdot \vec{n} dS, $$ 

where $\vec{F} (x,y,z) = x\ \vec{i} + y\ \vec{j} + z\ \vec{k}$.


\end{enumerate}







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