A manufacturing engineer is studying a family of solids whose bases lie on the closed…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A manufacturing engineer is studying a family of solids whose bases lie on the closed interval [0,L] along the x–axis. At each value of x, a cross section perpendicular to the x–axis is an identical geometric shape whose area can be written in the form A(x)=kg(x), where k>0 is a scale factor that depends on tooling settings and g is a fixed, positive function of x determined by the chosen base curve.
The total volume V of such a solid is given to be a constant that the finished product must meet exactly.
(a) Derive an expression for the volume V in terms of k, g, and L. Clearly state any conditions on the integrand that justify the steps of your derivation.
(b) Suppose the engineer doubles the scale factor k while keeping g fixed. Describe qualitatively how the required interval length L must change so that the volume remains exactly V. Support your answer with appropriate reasoning based on your result from part (a).
(c) Explain why the relationship you described in part (b) would still hold if the function g were multiplied by any positive constant factor, and discuss one practical implication of this fact for the design process.