Let be the region in the plane bounded on the right by the line
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the plane bounded on the right by the line x=y+2, on the left by the curve x=y2, and between y=0 and y=3. The region R is revolved about the vertical line x=−1.
Respond to the prompts without evaluating the resulting integrals.
(a) For a representative cross–section perpendicular to the axis of revolution, write formulas for the outer radius R(y) and inner radius r(y) in terms of y. Provide brief reasoning for each expression.
(b) On the interval 0≤y≤3, determine which function produces the larger radius at every value of y, and explain how this fact influences your integrand.
(c) Write the definite integral that represents the volume of the solid generated by revolving R about x=−1.
(d) Suppose instead the region were revolved about the line x=5. Without recomputing the entire integral, describe qualitatively how each radius expression would change.