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{\bf Math 222 in class problems} \hspace{.5in} \textbf{Week: March 1, 2021} %outlier malcom
\begin{enumerate}

%SPHERICAL AND CYLINDRICAL PROBLEMS
%DO CHANGE OF COORDINATES IN WEEK 7

%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.

\item  
Evaluate
\[
\displaystyle \int_{-2}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{\sqrt{x^2+y^2}}^2 x^2+y^2 \ dz \ dy \ dx
\]

\item  Find the volume of the region lying between the planes $z=0$ and $z=4-x$,
 outside the cylinder with base $r = 2\cos \theta$, and inside the cylinder with base $(x-2)^2 + y^2 =4$. 
 
\item Set up the integral representing the volume of the solid region bounded between $z=4,$\\ $z=1-x^2-y^2$, which is inside $x^2+y^2=1$.

 \item HW \#5 4b) \\
 Sketch the solid whose volume is given by the integral and evaluate the integral
\[
\int_0^{\pi/4}\int_0^{2\pi}\int_0^{\sec \varphi} \rho^2 \sin \varphi \  d\rho \ d\theta \ d\varphi
\]

\item Graph the region of integration and convert to spherical coordinates in order to evaluate (but do not do so):
\[
 \int_{0}^3 \int_{0}^{\sqrt{9-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{18-x^2-y^2}} x^2+y^2 + z^2 \ dz \ dx \ dy
\]


  
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