A company designs spinning toys using the family of functions $y = cx\sqrt{4 - x^{2}}$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = cx\sqrt{4 - x^{2}}$ for $c = 6$.
(b) It is known that, for $y = cx\sqrt{4 - x^{2}}$, $\frac{dy}{dx} = \frac{c(4 - 2x^{2})}{\sqrt{4 - x^{2}}}$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?
(c) For another spinning toy, the volume is $2\pi$ cubic inches. What is the value of $c$ for this spinning toy?

A company designs spinning toys using the family of functions $y = c x \sqrt { 4 - x ^ { 2 } }$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$ for $c = 6$.
(b) It is known that, for $y = c x \sqrt { 4 - x ^ { 2 } } , \frac { d y } { d x } = \frac { c \left( 4 - 2 x ^ { 2 } \right) } { \sqrt { 4 - x ^ { 2 } } }$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?
(c) For another spinning toy, the volume is $2 \pi$ cubic inches. What is the value of $c$ for this spinning toy?

When a solid of revolution is created by rotating the figure enclosed by the curve $y = a \left( 1 - x ^ { 2 } \right)$ and the $x$-axis around the $y$-axis, and the volume of the solid of revolution is $16 \pi$, find the positive value of $a$. [3 points]

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Let $V$ be the volume of the solid of revolution obtained by rotating $\Gamma$ about the $x$-axis. For all $a \in \left[-\frac{1}{2}, 1\right]$, is $V$ always equal? If equal, find its value; if not equal, find the value of $a$ for which $V$ has a maximum value, and find this maximum value. (Non-multiple choice question, 6 points)