2021 AP Calculus AB FRQ #3

A company designs spinning toys using the family of functions y = cx * (4 − x^2) ^ 1/2 , 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 * (4 − x^2) ^ 1/2, 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 ofy = cx * (4 − x^2) ^ 1/2 for c = 6.
(b) It is known that, for y = cx * (4 − x^2) ^ 1/2, dy/dx = (c * ( 4- x^ 2)) / (4 - x^2)^1/2 . 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?
Timestamps
Intro: 00:00
Part a: 00:32
Part b: 06:14
Part c: 09:25