MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A cubic curve is given by x=y3−3y. Let U be the closed region in the plane bounded by this curve and the y–axis between the two non-zero values of y where the curve intersects the axis.
(1) Solve y3−3y=0 to determine all y-values where the curve intersects the y–axis, and list them in ascending order.
(2) For each subinterval determined by the y-intercepts, decide whether the curve lies to the right (positive x) or to the left (negative x) of the y–axis. Justify your classification.
(3) Set up one or more definite integrals whose combined value equals the total area of region U. Clearly indicate all limits of integration and show how the sign of the integrand is addressed so that each integral contributes positively to the total area.
(4) Evaluate the integral expression(s) from part (c) to obtain the exact total area of U.