Let be the region in the first quadrant bounded above by the curve
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded above by the curve y=x, below by the x–axis, and on the right by the line x=4. A solid is formed by taking, at each value of x in [0,4], a rectangular cross-section perpendicular to the x–axis whose height is exactly twice the length of its base (the base lies entirely in R).
(a) Describe in words how the length of the base of a typical rectangular cross-section at position x is determined from the defining curves of R. Your description should make explicit reference to the region’s boundaries.
(b) Write, but do not evaluate, a definite integral that represents the volume of the solid.
(c) Explain why the integrand in part (b) involves a square of a function of x, even though the cross-sections are rectangles rather than squares.