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Problem $3$: $[20$ pts$]$ Avery wants to design a tank to produce the world's largest elliptical donut. They do so by taking the ellipse $16x^2+y^2=16$ and revolving it about the line $x=7.$ They want to fill the tank with donut batter.

A$.[2$ pts$]$ Use the internet to find a reasonable density for the donut batter ink$\frac{g}{m^3}.$ Cite your source.

According to

the density of the donut batter is

$k\frac{g}{m^3}.$

Unfortunately, Avery only has enough batter to fill the tank up to $y=3$ with donut batter. In order to avoid it from spoiling until they obtain more, Avery decides to drain all of the batter out of the bottom of the tank so it can be stored.

B$.[18$ pts$]$ Set up$,$ but do not evaluate, an integral or sum of integrals that gives the magnitude of the work required to drain all of the donut batter to the bottom of the tank. You should use the density you found and take $g=9\mathrm{.}8\frac{m}{s^2}.$ To earn full credit, include a labeled picture as part of your solution and show all of your work.

$11$

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Problem $2$: $[20$ pts$]$ The region $R$ shaded below is bounded by $x^2+y^2=8$ and $y=x^2.$

Exhibit an integral or sum of integrals that would represent the perimeter of the boundary of region $R.$

For each integral you write down, explain $($via a complete English sentence or by labeling the graph$)$ which part of the perimeter it computes.

If you utilize symmetry when finding the perimeter, make sure to explain how you used it$.$

Use technology to calculate the perimeter to $3$ decimal places.

You may utilize computational technology $($not AI$)$ to do this problem in any way you would like $($finding intersection points, derivatives, integrals, etc$).$ Make sure to state what you are computing, what you use to perform calculations, and report your final answer to $3$ decimal places.

$10$

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$($Problem $1,$ continued$)$

Recall that $R$ is bounded to the left by $x=0,$ below by $y=x^4,$ and above by $y=16.$

Part $2$: $15pts$

Now, suppose that $R$ is divided into two pieces by the line $y=b,$ where $0b16.$

Let $S_1$ denote the solid obtained when the part of $R$ that lies below $y=b$ is revolved about the $y-$axis

Let $S_2$ denote the solid obtained when the part of $R$ that lies above $y=b$ is revolved about the $y-$axis.

Suppose it is known that the volumes of $S_1$ and $S_2$ are equal.

A$.[3$ pts$]$ Without performing any calculations, explain whether you think that $b>$aabbbb, $orb>a,$ where $ais$ the value you obtained $in$ Part $1.$ Your response will $be$ graded $on$ the quality and completeness $of$ your explanation. You will NOT $be$ graded for correctness $on$ this part.

$B.[9pts]$ Calculate $b$ and show all $of$ your work. You may use technology $to$ solve the equation you obtain for $b,$ but you must evaluate all integrals $by$ hand. Make sure $to$ state what you use $to$ find $b.$

$C.[3pts]Is$ your result from Part $B$ consistent with your conjecture from Part A for this problem? $Ifitis$ not, provide a reflection $to$ resolve the discrepancy.

$(You$ may use the next page $to$ show additional work$)$

$8$

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Problem $1$: $30pts$ The region $R$ is bounded to the left by $x=0,$ below by $y=x^4,$ and above by $y=16.$

Part $1$: $[15$ pts$]$

Suppose that $R$ is divided into two pieces by the line $y=a,$ where $0a16.$ Let $R_1$ denote the part of $R$ that lies below $y=$a and $R_2$ be the part of $R$ that lies above $y=a.$ Given that the area of $R_1$ and the area of $R_2$ are equal, answer the following questions.

A$.[3$ pts$]$ Without performing any calculations, estimate the value of $a,$ and explain how you obtained it$.$ You should include a graph as part