Consider the region in the –plane that is bounded above by the line
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region T in the xy–plane that is bounded above by the line y=4−2x, bounded below by the curve y=x, and bounded on the left by the y–axis. The two graphs intersect at x=4. The region T is revolved about the horizontal line y=4, forming a solid.
(a) Verify algebraically that the two bounding graphs intersect when x=4, and state the corresponding y–coordinate of the point of intersection.
(b) Write, but do not evaluate, a definite integral whose value equals the volume of the solid.
(c) Evaluate the integral from part (b) to obtain the exact volume of the solid. Express your answer in terms of π. Show all analytical work that leads to your result.