A solid has a rectangular base lying in the –plane
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has a rectangular base lying in the xy–plane. The base is the region bounded by the x–axis and the curve y=g(x), where g is continuous on [a,c] and g(x)>0 for all x in that interval. For a≤x≤b, cross sections taken perpendicular to the x–axis are semicircles whose diameter lies in the base. For b≤x≤c, cross sections taken perpendicular to the x–axis are equilateral triangles whose side lies in the base.
(a) Write a single piecewise‐defined expression for the cross‐sectional area A(x) on the interval [a,c] in terms of g(x).
(b) Using your expression from part (a), formulate (but do not evaluate) an expression for the exact volume V of the solid as a single definite integral on [a,c]. Clearly indicate the limits of integration.
(c) Explain, without performing any integration, why the expression in part (b) represents the total volume of the solid even though the cross‐sectional shape changes at x=b. In your explanation, explicitly reference the fundamental properties of definite integrals that justify combining the two regions into one integral.
(d) Suppose g is differentiable on (a,c) but g′(x) has a jump discontinuity at x=b. Discuss whether this affects the validity of the volume expression derived in part (b), and justify your conclusion.