Let be the region enclosed by the graphs of and on the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let T be the region enclosed by the graphs of y=x and y=x2 on the interval 0≤x≤4. The region is revolved about the horizontal line y=1.
(1) Find all x–coordinates in [0,4] at which the line y=1 intersects either y=x or y=x2. Show your work.
(2) Using the intersections from part (a), express the volume of the resulting solid as the sum of one or more definite integrals. Each integrand should be written entirely in terms of x and should correctly reflect which curve is closer to and which is farther from the axis of rotation on each subinterval.
(3) Evaluate every integral in part (b) and compute the exact volume of the solid. Provide your final answer in simplest form including π. Clearly indicate any algebra used to simplify the result.