Region P is bounded above by the horizontal line and below by two different lines
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region P is bounded above by the horizontal line y=6 and below by two different lines, forming two adjacent portions along the x-axis:
• For 0≤x≤3, the lower boundary is y=2x.
• For 3≤x≤4, the lower boundary is y=12−2x.
A solid has P as its base. At every x between 0 and 4, a cross section perpendicular to the x-axis is a square whose base lies in P. Carry out the following tasks.
(a) Determine the length s1(x) of the side of a square cross section for 0≤x≤3.
(b) Determine the length s2(x) of the side of a square cross section for 3≤x≤4.
(c) Write definite integrals that represent the volumes contributed by each portion of the base, and evaluate both integrals.
(d) State the total volume of the solid, including units, and briefly comment on how the change in the lower boundary at x=3 affects the integration process.