Consider the region in the –plane satisfying the following conditions

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Consider the region TT in the xyxy–plane satisfying the following conditions. • For 0≤x≤20\le x\le 2 the upper boundary is y=6−x2.y=6-x^{2}. • For 2≤x≤42\le x\le 4 the upper boundary is y=2x+1.y=2x+1. • For all 0≤x≤40\le x\le 4 the lower boundary is the horizontal line y=1.y=1. The region TT is revolved about the horizontal line y=4y=4 to form a solid. (a) Verify algebraically that the two upper-boundary formulas meet at (2,2⋅2+1).(2,2\cdot2+1). State the yy–coordinate of this point and explain why continuity of the boundary at x=2x=2 matters for volume computation. (b) Set up, but do not evaluate, the exact expression for the volume of the solid. Be sure to partition the domain appropriately and label all radii in terms of x.x. (c) Evaluate the integral(s) from part (b) to obtain the exact volume.

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