Let be the region enclosed by the parabola and the -axis
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let T be the region enclosed by the parabola y=4−x2 and the x-axis. A solid is generated by taking cross-sections perpendicular to the x-axis; each cross-section is an isosceles right triangle whose legs rest on the base region at that x-value.
Work through the following tasks carefully and present a logically organized solution.
(a) Determine the exact x-coordinates where the parabola intersects the x-axis, and state the resulting limits of integration.
(b) Express the length of a leg of a typical triangular cross-section as an explicit function of x.
(c) Derive a formula for the area of the cross-section in terms of x, and formulate the definite integral whose value equals the volume of the solid.
(d) Evaluate the integral found in part (c) to obtain the exact volume.