Consider the region in the –plane bounded on the right by the curve and on the left…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region in the xy–plane bounded on the right by the curve x=y+2 and on the left by the curve x=y2 for 0≤y≤3. The entire region is revolved about the vertical line x=2 to create a three–dimensional solid.
(a) Calculate the y–coordinates of the points where the two bounding curves intersect.
(b) Decide whether the line x=2 lies entirely outside the region, entirely inside the region, or intersects the region itself. Provide a rigorous justification that references the two bounding curves.
(c) Explain, without computing any integral, whether the volume of the solid must be obtained with a single definite integral or with more than one definite integral. Support your conclusion with geometric reasoning.
(d) Describe qualitatively how the radius of each circular cross-section changes as y increases from the smallest to the largest intersection value found in part (a).