# The base of a solid is the region in the first quadrant enclosed by the graph of #y= 2-(x^2)# and the coordinate axes. If every cross section of the solid perpendicular to the y-axis is a square, how do you find the volume of the solid?

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To find the volume of the solid, we need to integrate the area of each cross-section perpendicular to the y-axis over the range of y-values that the solid occupies.

Since each cross-section perpendicular to the y-axis is a square, the area of each cross-section will be equal to the square of its side length. We need to find an expression for this side length in terms of y.

Consider a typical cross-section at a height y. This cross-section will have two sides parallel to the x-axis and two sides parallel to the y-axis. Since the cross-section is a square, these sides will be equal.

The distance between the y-axis and the curve y = 2 - x^2 at height y is the x-coordinate of the point where the curve intersects the line y = y. Solving 2 - x^2 = y for x, we get x = ±√(2 - y). Since we're only interested in the region in the first quadrant, we consider x = √(2 - y).

Therefore, the side length of the square cross-section at height y is 2√(2 - y).

Now, the area of this square cross-section is the square of its side length, which is (2√(2 - y))^2 = 4(2 - y).

To find the volume of the solid, integrate the area of each cross-section over the range of y-values from 0 to the y-value where the curve y = 2 - x^2 intersects the y-axis.

The curve intersects the y-axis when x = 0, so 2 - x^2 = 0, which gives y = 2. Therefore, the limits of integration are from y = 0 to y = 2.

The volume (V) of the solid is given by the integral:

[V = \int_{0}^{2} 4(2 - y) , dy.]

Evaluating this integral gives the volume of the solid.

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