Let be the subset of the upper half of the circle that lies to the right of the…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the subset of the upper half of the circle x2+y2=9 that lies to the right of the vertical line x=2 (y≥0 and x≥2). The region R is revolved about the y-axis, creating a solid with a cylindrical void.
(a) Express the outer radius and the inner radius of a typical cross-sectional circular slice of the solid at height y as explicit functions of y.
(b) Determine the exact y-interval over which these radii are defined for the solid.
(c) Write a definite integral that represents the volume of the solid, making sure to use the limits found in part (b).
(d) Evaluate the integral from part (c) to obtain the exact volume of the solid in terms of π. Show all algebraic steps leading to your final answer.