Problem Setup
Base: The region in the xy-plane bounded by:
• y = x1/2 = √x
• y = 0 (x-axis)
• x = 4 (vertical line)
Cross-sections: Perpendicular to the x-axis, each cross-section is a square.
Key Insights
At any x-value between 0 and 4:
• The height of the base region is: h(x) = √x
• Since cross-sections are squares, the side length = √x
• The square extends from y = 0 to y = √x in the xy-plane
• The square extends from z = 0 to z = √x in the z-direction
Volume Calculation
V = ∫₀⁴ [side length]² dx = ∫₀⁴ (√x)² dx = ∫₀⁴ x dx = [x²/2]₀⁴ = 8 cubic units