Consider the region bounded by the curves and between and
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region T bounded by the curves y=x and y=x2 between x=0 and x=3. The region is revolved about the horizontal line y=1, which lies between the two curves for some portion of the interval.
(a) Determine all values of x in [0,3] at which either curve intersects the line y=1. Explain how these values will affect the setup of any integral used to compute the volume of the solid generated.
(b) Write a piecewise–defined definite integral (or sum of definite integrals) that represents the exact volume of the solid obtained by the rotation of T about y=1. Clearly indicate the intervals over which each expression applies.
(c) Evaluate your integral(s) from part (b) to obtain the exact volume.