Let R be the region bounded by the graphs of f(x)=√x and g(x)=x/2?
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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√(x) = x/2
x= x^2/4
4x = x^2 . . . . .x^2 - 4x = 0 . . . .x(x - 4) = 0 . . . .x = 0 & 4
4
∫ ( Area of square ) dx
0
4
∫ b^2 dx
0
4
∫ ( √(x) - x/2 )^2 dx
0
4
∫ (x - x√(x) + x^2/4 ) dx
0
4
∫ (x - x^(3/2) + x^2/4 ) dx
0
. . . . . .. . ...... . .. . . .. . . .. . .. . . .. .. . .. . . ... .. . .4
(1/2) * x^2 - x^(3/2 + 1)/(3/2 + 1) + (1/4) * (1/3) * x^3 ]
. . . . . .. . ...... . .. . . .. . . .. . .. . . .. .. . .. . . ... .. . .0
(1/2) * (4^2 - 0^2) - (2/5) * ( 4^(5/2) - 0^(5/2) ) + (1/12) * ( 4^3 0^3 )
(1/2) * (16 - 0) - (2/5) * ( 32 - 0 ) + (1/12) * ( 64 - 0 )
8 - (64/5) + (64/12)
8/15
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