Let be the region in the first quadrant bounded by the curves ,
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded by the curves y=x2, y=2x, and the y–axis. The region R is revolved about the vertical line x=4 to form a solid.
(a) Write a single definite integral with respect to x whose value equals the volume of the solid. Include explicit expressions for the outer and inner radii.
(b) Rewrite the volume as a single definite integral with respect to y. Show any algebraic work needed to express the radii purely in terms of y. Do not evaluate at this stage.
(c) Evaluate either integral from parts (a) or (b) to obtain the exact volume of the solid. Then verify algebraically that the other integral would yield the same numerical value if evaluated.