How to find volume between two functions with cross sections

Question

Problem: What is the volume of the solid whose base is the region in the first quadrant bounded by and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are squares?

Answer 1

Intersection: 2 - x = √x; 4 - 4x + x² = x; x² - 5x + 4 = 0; x = 1 or x = 4 . we have to reject x = 4

V = ∫₀¹(2 - x - √x)²dx = ∫₀¹(4 + x² + x - 4x - 4√x + 2x√x)dx = (4x + x³/3 - 3x²/2 - 8/3x^3/2 + 4/5x^5/2)₀¹ =

4 + 1/3 - 3/2 - 8/3 + 4/5 = 29/30