The base of a solid region in the first quadrant is bounded by the x-axis,y-axis, the graph of $y = x^{2} + 1$ $y=x^2+1$ , 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

$= \frac{206}{15}$ $= 206/15$

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

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

$A (x) = {(x^{2} + 1)}^{2}$ $A(x) = ( x^2 + 1)^2$

thus the volume is

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

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

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

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

$= 206/15$

The base of a solid region in the first quadrant is bounded by the x-axis,y-axis, the graph of y=x^2+1y=x2+1y=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?