Let be the region in the first quadrant bounded by the curves and between their first…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let P be the region in the first quadrant bounded by the curves y=sinx and y=cosx between their first two points of intersection. Rectangular cross-sections are taken perpendicular to the x–axis so that the height of each rectangle is four times the length of its base, which lies entirely in P.
(a) Determine all x–coordinates where sinx=cosx in the first quadrant and justify that these are the only points of intersection relevant for this problem.
(b) For x in the interval found in part (a), express the length of the base of a cross-section as a function of x.
(c) Set up a definite integral (or integrals, if necessary) that gives the volume of the solid.
(d) Evaluate the integral(s) to obtain an exact expression for the volume.
(e) The solid is to be re-scaled so that its volume equals 10 cubic units while preserving the same cross-sectional proportions throughout. Determine the constant factor by which every linear dimension of the original solid must be multiplied and explain your reasoning.