Problem $1$: $[30$ pts$]$ 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 $0$ a $16.$ 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 of your explanation. Your response will be graded on the quality and

completeness of your explanation. You will NOT be graded for correctness on this part.

B$.[9$ pts$]$ Calculate a and show all of your work. You may use technology to solve the equation you obtain for a$,$

but you must evaluate all integrals by hand. Make sure to state what you use to find a$.$

C$.[3$ pts$]$ Is your result from Part B consistent with your result from Part A$?$ If it is not, provide a reflection to

resolve the discrepancy

$($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 $2$: $[15$ pts$]$

Now, suppose that R is divided into two pieces by the line y $=$ b$,$ where $0$ b $16.$

$$ 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 $<$ a$,$ b $=$ a$,$ or b $>$ a$,$ where a is

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$.[9$ pts$]$ 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$.[3$ pts$]$ Is your result from Part B consistent with your conjecture from Part A for this problem? If it is not,

provide a reflection to resolve the discrepancy.