Question

Region R is bounded by the curves y = √x, y = 1, and x = 4. A solid has base R, and cross sections perpendicular to the x-axis are squares. The volume of this solid is

A. 4/3
B. 8
C. 7/6
D. 15/2

Answer 1

The cross sections have side length equal to the vertical distance between y = √x and y = 1, or |√x - 1|. The two curves meet at the point (1, 1), and y = √x meets x = 4 at (4, 2), so we'll be integrating with respect to x on the interval [1, 4]. Over this interval, √x ≥ 1, so |√x - 1| = √x - 1.

A cross section of thickness ∆x has volume

(√x - 1)² ∆x = (x - 2√x + 1) ∆x

Then the volume of the solid is

`\displaystyle \int_1^4 (x - 2\sqrt x + 1) \, dx = \boxed{\frac76}`

Answer 2

the answer is B. hope this helped