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 diagonals run from the semicircle $y=-\sqrt{1-x^{2}}$ to the semicircle $y=\sqrt{1-x^{2}}$.

Key Concepts

Cross-Sectional Slicing

This technique involves breaking a solid into a series of planar slices perpendicular to a chosen axis. By determining the area of each slice as a function of position along that axis, one may integrate these areas over the interval to obtain the volume of the solid.

Relationship Between a Square's Diagonal and Its Side Length

The side length of a square is directly related to the length of its diagonal according to the formula side = (diagonal/?2). Understanding this relationship is crucial when the diagonal is determined by other geometric constraints, as it allows one to compute the square's area in terms of the given diagonal.

Function Boundaries in Geometric Regions

Using functions to describe the boundaries of regions allows for precise definitions of areas and volumes. In many problems, curves such as those defined by square root functions delineate regions of interest, and these boundaries are used to set the limits for integration or to determine dimensions of cross-sectional areas.

Integration for Volume Calculation

Integration is a fundamental process in calculus that sums up an infinite number of infinitesimally small quantities. When applied to solids with known cross-sectional areas, integration along the axis that the slices are taken from yields the entire volume of the solid.

Transcript

00:01 In this question, we're asked to find the volume of a region that's bound between two semicircles, two semicircles that form a complete circle when we put them together.

00:15 So we have y equals the square root of 1 minus x squared, and we have y equals negative the square root of 1 minus x squared.

00:28 We want to find the volume of a region with cross sections that are squares, so that the diagonal lies in this plane between those two.

00:36 Two curves.

00:38 So that's the diagonal of one of our squares.

00:41 And we want to do it from negative one here to one on this side where the two circles meet.

00:49 Now, in order to find this volume, we have to integrate over the area of those squares, and the area of a square is based on the side of the square.

00:57 So we have to get from the diagonals to the sides.

00:59 Now, the diagonal of a square is radical 2 times its side length, meaning that if we solve that for the side length, we get the side length is the diagonal divided by radical 2.

01:15 And here, the diagonal of these squares, since it's bounded between these two curves, is the difference between them.

01:25 So our side length here is going to be the difference of those two curves, radical 1 minus x squared minus negative radical 1 minus x squared over radical 2.

01:39 Now we can do a little bit of manipulation here to make this easier to work with.

01:45 First of all, that double negative there can become addition, and then we add those two together, and we get two times the square root of 1 minus x squared over the square root of 2.

01:56 Next, if we rationalize this, if we multiply by radical 2 on the top and bottom, we get 2 square root of that radical 2 can multiply into the other radical.

02:10 So 2 minus 2x squared over 2.

02:14 And then those twos cancel out.

02:16 So we get the side length of the square...

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