Let be a solid whose base region in the -plane is the triangle with vertices ,
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let T be a solid whose base region in the xy-plane is the triangle with vertices (0,0), (6,0), and (0,6). For x in [0,6] the cross-sectional shape taken perpendicular to the x-axis is determined by the following piecewise rule:
• If 0≤x<2.5, the cross section is a rectangle whose height is 23 times its base.
• If 2.5≤x≤4, the cross section is a square.
• If 4<x≤6, the cross section is a rectangle whose height equals one half of its base.
The solid is carved such that the change from one type of cross section to the next occurs sharply at x=2.5 and x=4 with no gaps or overlaps.
(a) For x in [0,6], express the base length of a cross section in terms of x by writing it as a function b(x) using the equation of the line that forms the hypotenuse of the triangular base.
(b) Using the information provided, write a fully simplified piecewise formula for the cross-sectional area A(x) on 0≤x≤6.
(c) Determine whether A(x) is continuous at x=2.5 and x=4. Provide justification for each point.
(d) Evaluate the appropriate definite integral to obtain the exact volume of T.