A solid has a base in the first quadrant bounded above by the parabola and below by…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has a base in the first quadrant bounded above by the parabola y=x2 and below by the x-axis for 0≤x≤2. Cross–sections taken perpendicular to the x-axis are semicircles whose diameters lie on the base region.
Respond to each part thoroughly, using correct mathematical reasoning and notation.
(a) Express the diameter of a typical semicircular cross-section as a function of x. Clearly state any domain restrictions.
(b) Using your result from part (a), write an explicit algebraic expression for the area A(x) of the semicircular cross-section at position x.
(c) Write—but do not evaluate—a definite integral whose value equals the volume of the solid. Explain how the limits of integration are determined from the geometry of the base region.