A solid has base region S in the first quadrant bounded below by the curve and above…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has base region S in the first quadrant bounded below by the curve y=x2 and above by the line y=2x. For every x in the interval 0≤x≤2, a cross section of the solid taken perpendicular to the x-axis is constructed on S.
• When 0≤x≤1, each cross section is a square whose base sits entirely in S.
• When 1≤x≤2, each cross section is a rectangle whose height is exactly twice the length of its base.
Answer the following, showing all steps and justifying each conclusion.
(1) Part 1: (a) Determine the x-coordinates where the bounding curves y=x2 and y=2x intersect inside the first quadrant.
(2) Part 2: (b) For 0≤x≤1, express the length b1(x) of the base of the square cross section as an explicit function of x.
(3) Part 3: (c) Write, then evaluate, a definite integral representing the volume V1 of the portion of the solid formed by the square cross sections.
(4) Part 4: (d) For 1≤x≤2, express the base length b2(x) and the corresponding height h2(x) of the rectangular cross section as explicit functions of x.
(5) Part 5: (e) Write, then evaluate, a definite integral representing the volume V2 of the portion of the solid formed by the rectangular cross sections. Hence determine the total volume of the solid, giving units of volume.