MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region S is enclosed by the curves y=x and y=x2.
(1) Find the coordinates of all intersection points of the two curves.
(2) Set up and evaluate definite integral(s) that give the area of S by integrating with respect to x. Show all work, including limits of integration.
(3) Rewrite each curve so that x is expressed as a function of y. Then set up and evaluate a single definite integral that gives the area of S by integrating with respect to y.
(4) Compare the algebraic complexity of the two computations in parts (b) and (c). Cite specific features of the integrands or antiderivatives that support your comparison.
(5) Verify that the numerical values obtained in parts (b) and (c) are equal, and explain why this must be the case for the region S."