# 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?

thus the volume is

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To find the volume of the solid, integrate the area of each square cross-section perpendicular to the x-axis over the given interval.

The side length of each square cross-section is determined by the difference between the y-values of the bounding curves along the x-axis. In this case, the bounding curves are the x-axis and the curve y = x^2 + 1.

To find the side length of each square, subtract the y-coordinate of the curve from the y-coordinate of the x-axis. Therefore, the side length of each square is given by: s = (x^2 + 1) - 0 = x^2 + 1.

The volume, V, of the solid is the integral of the area of each square cross-section with respect to x over the interval [0, 2]:

V = ∫[0 to 2] (x^2 + 1)^2 dx.

Now, integrate the function (x^2 + 1)^2 with respect to x over the interval [0, 2] to find the volume of the solid.

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