
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\g{\bigtriangledown}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%end macros
 
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{\bf NAME:} 
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{\bf Section:} 
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{\bf MATH 251 (1-6,10-11),  Dr. Z. ,  Fourth Practice Test for Exam \#2}

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1. \qquad  (out of 10)    
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2.  \qquad  (out of 10)    
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3.   \qquad (out of 10)    
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4.   \qquad (out of 10)    
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6.    \qquad (out of 10)    
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8.    \qquad (out of 10)    
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9.    \qquad (out of 10)    
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10.    \qquad (out of 10)    
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{\bf total}:
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{\bf 1.} (10 points)
Find the maximum of $xyz$ subject to the constraint $x+y+z=6$.

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{\bf Ans:}
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{\bf 2.} (10 points)
Evaluate the line integral
$$
\int_{C} 2y\,dx+ 3x \, dy + 6yz \,dz \quad ,
$$
where $C: x=t^2 \, , \,  y=t^2 \, , \, z=2t \, , \, 0 \leq t \leq 2$.

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{\bf Ans.:} 
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{\bf 3.} (10 points)
Evaluate the line integral,
$$
\int_{C} x^2 y \, ds \quad,
$$
where $C$ is top half of the circle $x^2+y^2=9$.
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{\bf Ans.:} 
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{\bf 4.} (10 points)
Evaluate the triple integral
$$
\int\int\int_{E} \,\,  \, yz e^{x^5} \, dV \quad ,
$$
where
$$
E=\{ (x,y,z) \, \vert \, 0 \leq x \leq 1 \, , \, 0 \leq y \leq 2x
\, ,  \, 2x \leq z \leq 3x \, \} \quad .
$$
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{\bf Ans.:} 
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{\bf 5.} (10 points)
Evaluate the integral
$$
\int\int_{D} e^{-x^2-y^2} \, dA \quad,
$$
where 
$$
D=\{ (x,y) \, \vert \, \, \, \, x \geq y \geq 0 \, \, , \, \,  x^2+y^2 \leq 1 \} \quad .
$$

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{\bf Ans.:} 
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{\bf 6.} (10 points) Evaluate the integral
$$
\int\int_{D} \,\, \, e^{-4x^2-4y^2} \, dA \quad,
$$
where $D$ is the region bounded by the semi-circle
$x=\sqrt{1-y^2}$ and the $y$-axis.
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{\bf Ans.:} 
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{\bf 7.} (10 points)
Evaluate the triple integral
$$
\int\int\int_{E} yz \sqrt{1+x^5} \, dV \quad ,
$$
where
$$
E=\{ (x,y,z) \, \vert \, 0 \leq x \leq 1 \, , \, 0 \leq y \leq x
\, ,  \, x \leq z \leq 2x \, \} \quad .
$$

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{\bf Ans.:} 
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{\bf 8.} (10 points) Use Lagrange multipliers to find the maximum and
minimum values of $f(x,y)=1+3x+2y$ subject to the
constraint $x^2+y^2=25$.
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{\bf maximum value:} 
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{\bf minimum value:} 
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{\bf 9.}  (10 points)
Find the volume of the solid bounded by the cylinder $y=x^2$
and the planes $z=0$ and $y+z=1$. Simplify as much as you can.

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{\bf Ans.:}
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{\bf 10.} (10 points)
Sketch the region of integration
and change the order of integration.
$$
\int_{0}^{2} \int_{x^3}^{8} g(x,y) \, dy \, dx
$$
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{\bf Ans.:} 
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