MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region W is bounded above by y=2, below by y=(x−1)2, and on the right by x=4. The region is revolved about the horizontal line y=5 to create a solid.
(a) Determine the x-coordinate(s) where the curve y=(x−1)2 intersects the line y=2 within the interval 0≤x≤4.
(b) Justify that these intersection points partition the region into subregions on which the distance from the axis of rotation to the lower boundary changes functional form.
(c) Write a sum of definite integrals, each with respect to x, that gives the volume of the solid. Clearly indicate the limits of integration for each integral.
(d) Evaluate the integrals to find the exact volume of the solid.