A region in the first quadrant is bounded above by the line
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region R in the first quadrant is bounded above by the line y=2x, below by the parabola y=x2, and on the left by the y–axis. The region is revolved about the horizontal line y=3.
(a) Sketch or describe the key features of R, including all intersection points used to determine its horizontal extent.
(b) For a representative value of x in the interval defining R, express both the outer radius R(x) and the inner radius r(x) of the resulting circular cross–section created by the rotation. Be explicit about the domain on which each expression is valid.
(c) Explain whether the expressions found in part (b) remain valid over the entire interval of integration or whether the interval must be subdivided. Provide a clear justification based on the relative positions of the curves and the line of rotation.