Consider the region in the -plane enclosed by the right-opening parabola and the…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region in the xy-plane enclosed by the right-opening parabola x=y2 and the left-opening parabola x=8−y2. A solid is formed whose cross sections perpendicular to the y-axis are isosceles right triangles with legs lying in the region and hypotenuse perpendicular to the y-axis.
Coordinate units are meters.
(a) Determine the y-coordinates of all intersection points of the two parabolas. Show algebraic justification.
(b) For each y between the intersection points, express the length ℓ(y) of a leg of the triangular cross section in terms of y. Simplify completely.
(c) Write a single definite integral that equals the volume V of the solid. Clearly identify the integrand and the limits of integration.
(d) Evaluate the integral from part (c) to find an exact value for V.
(e) Without performing additional computation, explain whether replacing the isosceles right triangles with squares (still using the same base region and orientation) would increase, decrease, or leave unchanged the volume of the resulting solid. Support your conclusion with geometric reasoning.