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%	\mbox{}
%	\title{Vector Calculus (MATH 53)}
%	
%	\maketitle
	
	
\section*{Double Integrals}

\begin{problem}[Stewart, Exercise~15.1.11]
	Evaluate the double integral
	\[
		\iint_R (4-2y) \, dA, \quad R = [0,2] \times [0,1],
	\]
	by first identifying it as the volume of a solid.
\end{problem}

\begin{problem}[Stewart, Exercise~15.1.34]
	Evaluate the double integral
	\[
		\iint_R \frac 1{1 + x + y} \, dA, \quad R = [1,3] \times [1,2].
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.1.42]
	Find the volume of the solid in the first octant bounded by the cylinder $z = 16 - x^2$ and the plane $y=5$.
\end{problem}

%\begin{problem}[Stewart, Exercise~15.1.43]
%	Find the volume of the solid enclosed by the paraboloid $z = 2 + x^2 + (y-2)^2$ and the planes $z = 1$, $x = 1$, $x = -1$, $y = 0$, and $y = 4$.
%\end{problem}

\begin{problem}[Stewart, Exercise~15.1.49]
	Use symmetry to evaluate the double integral
	\[
		\iint_R \frac {xy}{1 + x^4} \, dA, \quad R = [-1,1] \times [0,1].
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.22]
	Evaluate the double integral
	\[
		\iint_D y \, dA,
	\]
	where $D$ is the triangular region with vertices $(0,0)$, $(1,1)$, and $(4,0)$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.27]
	Find the volume of the tetrahedron enclosed by the coordinate planes and the plane $2x + y + z = 4$.
\end{problem}

\begin{problem}[Stewart, Example~15.2.4]
	Find the volume of the tetrahedron bounded by the planes $x + 2y + z = 2$, $x=2y$, and $z=0$ using:
	\begin{enumerate}
		\item double integrals,
		\item triple integrals.
	\end{enumerate}
	[Hint: The tetrahedron lies in the first octant. Sketch it by looking from your left.]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.40]
	Sketch the solid whose volume is given by the integral
	\[
		\int_0^1 \int_0^{1-x^2} (1-x) \, dy dx.
	\]
\end{problem}
\newpage

\begin{problem}[Stewart, Exercise~15.2.(51,52,55)]
	Evaluate the following integrals:
	\begin{enumerate}
		\item $\int_0^1 \int_{3y}^3 e^{x^2} \, dx dy$,
		\vspace{4pt}
		\item $\int_0^1 \int_{x^2}^1 \sqrt{y} \sin y \, dy dx$,
		\vspace{4pt}
		\item $\int_0^1 \int_{\arcsin y}^{\pi/2} \cos x \sqrt{1 + \cos^2x} \, dx dy$.
	\end{enumerate}
	[Hint: Reverse the order of integration first.]
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Calculate the iterated integral
	\[
		\int_0^4 \int_{\sqrt{x}}^2 e^{y^3} \, dydx.
	\]
\end{problem}

\begin{problem}[Stewart, Exercise~15.2.(65,67,68)]
	Use geometry to evaluate the following double integrals.
	\begin{enumerate}
		\item $\iint_D (x+2) \, dA$, \quad $D = \{(x,y) \mid 0 \le y \le \sqrt{9 - x^2}\}$,
		\vspace{5pt}
		\item $\iint_D (2x + 3y) \, dA$, \quad $D = [0,a] \times [0,b]$,
		\vspace{5pt}
		\item $\iint_D (2 + x^2y^3 - y^2 \sin x) \, dA$, \quad $D = \{(x,y) \mid |x| + |y| \le 1 \}$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercise~15.4.9]
	Find the mass and center of mass of the lamina that occupies the region bounded by the curves $y=e^{-x}$, $y=0$, $x=0$, and $x=1$ if its density function is $\rho(x,y) = xy$.
\end{problem}

\begin{problem}[Stewart, Exercise~15.5.3]
	Use the formula
	\[
		A(S) = \iint_D \sqrt{1 + f_x^2 + f_y^2} \, dA
	\]
	to find the surface area of the part of the plane $3x + 2y + z = 6$ that lies in the first octant.
\end{problem}

\begin{problem}[Stewart, Exercise~15.5.7]
	Find the surface area of the part of the hyperbolic paraboloid $z = y^2 - x^2$ that lies between the cylinders $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$. [Hint: Use the formula for surface area via double integrals.]
\end{problem}


\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}