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\line{Math 174\hfill Exam III\hfill April 9, 1999\hfill D.~Gottlieb}
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\item{1.}a)\ Give the definition of the triple integral of a function $f(x,y,z)$ over a bounded\newline
\null\quad region of space.

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\noindent
OR
\itemitem{b)}Evaluate $\ds{\int\limits_1^3\int\limits_1^3\int\limits_1^3\ {1\over xyz}\ dx dy dz}$.

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\item{2.}a)\ Find the average height of the hemisphere $z=\sqrt{a^2-x^2-y^2}$ over the disk\newline
\null\quad $x^2+y^2\leq a$.

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\noindent
OR

\itemitem{b)}The average value of function $f(x,y)$ equals $3\pi r$ over a disk $D$ of radius $r$ in the plane.
What is the value of $\ds\iint\limits_D f(x,y) dA$?

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\item{3.}The area of a region $R=\ds\int\limits_0^1\int\limits_{1-x}^{1-x^2}dy dx$.
Sketch $R$ and set up the integral of the area of $R$ with the order of integration reversed.

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\item{4.}Set up the triple integrals for the volume of the sphere of radius 5 and center at the origin using
\itemitem{a)}rectangular coordinates
\itemitem{b)}cylindrical coordinates
\itemitem{c)}spherical coordinates
\itemitem{d)}By the way, what is the volume of that sphere?
(You do not have to integrate if you already know the answer).

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\item{5.}a)\ Solve the system of equations\newline
$u=3x+2y,\ v=x+4y$ for $x$ and $y$ in terms of $u$ and $v$.
Then find the Jacobian $\partial(x,y)/\partial(u,v)$.

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\itemitem{b)}Let $ABC$ be a triangle in the $xy$ plane where $A=(0,0),\ B=(1,0),\ C=(0,1)$.
Let $T\colon (xy$--plane) $\longrightarrow (uv$--plane) be the transformation $T$ given by\newline
$(x,y)=(3x+2y,\ x+4y)$.
What is the image of the triangle under $T$ in the $uv$ plane?

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\item{6.}Consider the $L$ shaped region of width 1 in.

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\itemitem{a)}Find its centroid $(\overline x,\overline y)$.

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\noindent
OR

\itemitem{b)}Find its center of mass $\vec c$ when the $L$ has constant density by using Pappus's formula $\vec c=\ds{1\over m_1+m_2}\ (\vec c_1+\vec c_2)$ where $\vec c_1$ is the center of mass of a body of mass $m_1$ and $\vec c_2$ is the center of mass of a body of mass $m_2$ which does not overlap with body 1.
(You are not forced to integrate in this problem.)

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\bye
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\item{7.}a)\ Which function $f(x,y,z)=x^2+y^2+z^2$, or $g(x,y,z)=x+y+z$ has the fastest rate of increase at $(1,-1,1)$.

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AND

\itemitem{b)}What is the tangent vector to the curve of intersection at $(1,-1,1)$?

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\item{8.}At what point is the maximum value of $g(x,y,z)=x+y+z$ subject to the constraint $f(x,y,z)=x^2+y^2+z^2=3$ taken on?

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\item{9.}Find the equation of the tangent plane to the sphere $x^2+y^2+z^2=\ds{1\over 3}$ at the point $\ds{\bigg({1\over 3}, {1\over 3}, {1\over 3}\bigg)}$.

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\item{10.}Use Taylor's formula for $f(x,y)=e^x\cos y$ to find a quadratic approximation near the origin.

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\bye
