MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region T is bounded on the left by the line x=y−3 for −2≤y≤4. Its right-hand boundary changes as follows:
• For −2≤y≤1, the boundary is the curve x=5−y2.
• For 1≤y≤3, the boundary is the line x=7−y.
• For 3≤y≤4, the boundary is the line x=13−2y.
Assume all given intersections are actual points of intersection of the specified curves.
(a) Confirm analytically that, on each sub-interval of y, the right-hand boundary lies to the right of the left-hand boundary.
(b) Write a sum of definite integrals that represents the exact area of T.
(c) Evaluate the integrals obtained in part (b) to find the exact area of T.
(d) Suppose the density of a thin lamina occupying T is given by ρ(y)=3+siny (mass units per unit area). Set up, but do not evaluate, an expression for the total mass of the lamina.