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?
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?