The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares extending from y=f(x) to y=g(x). Write and evaluate the integral expression that can be used to find the volume of the solid.
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Then the base of each square is b = √x - x/2.
Then the area of each square is A = b² = x - x√x + x²/4
Then the volume is V = ∫[0,4] (x - x√x + x²/4) dx = (x²/2 - (2/5)x^(5/2) + x³/12) |[0,4]
V = 8 - (2/5)32 + 64/12 = (1/60)(480 - 768 + 320) = 32/60 = 8/15