The base of a solid region in the first quadrant is bounded by the x-axis,y-axis, the graph of #y=x^2+1#, and the vertical line x=2. If the cross sections perpendicular to the x-axis are squares, what is the volume of the solid?

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Answer 1 · 2016-06-24T20:27:11

#= 206/15#

the square cross section has sides of length #y = (x^2 + 1)#

so at any point #x#, the cross sectional area A(x) is:

#A(x) = ( x^2 + 1)^2#

thus the volume is

#V = int_0^2 \ A(x) \ dx = int_0^2 \ ( x^2 + 1)^2 \ dx#

#= int_0^2 \ x^4 + 2 x^2 + 1 \ dx#

#= [ x^5/5 + 2/3 x^3 + x ]_0^2#

#= [ 2^5/5 + 2/3 2^3 + 2 ]_0^2#

#= 206/15#