The solid lies between planes perpendicular to the x -axis...

The solid lies between planes perpendicular to the $x$ -axis at
$x=-1$ and $x=1 .$ The cross-sections perpendicular to the
$x$ -axis between these planes are squares whose bases run from the
semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$

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Key Concepts

Method of Slicing

The method of slicing involves dividing a solid into a series of thin cross-sectional slabs whose areas can be easily calculated. By integrating these areas along the axis perpendicular to the slices, the total volume of the solid can be determined. This technique is foundational in calculus for finding volumes of solids with non-uniform shapes.

Cross?Sectional Area Function

For solids with known cross-sectional shapes, an expression for the area of each slice as a function of position along the axis of integration is derived. This function reflects how the area of the slice changes from one end of the solid to the other and is essential for setting up the integral that represents the total volume.

Definite Integral Setup for Volume

Once the area function is obtained, the volume of the solid is computed by integrating this function over the interval corresponding to the extent of the solid along the chosen axis. The definite integral sums the contributions of each infinitesimally thin slice, resulting in an exact calculation of the volume.

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