Let B be the solid whose base is the unit circle x^2+y^2=1...

Let $B$ be the solid whose base is the unit circle $x^{2}+y^{2}=1$ and whose vertical cross sections perpendicular to the $x$-axis are equilateral triangles. Show that the vertical cross sections have area $A(x)=\sqrt{3}\left(1-x^{2}\right)$ and compute the volume of $B$.

Key Concepts

Variable Cross Sections

In solid geometry, a solid with variable cross sections is one whose slices perpendicular to a fixed axis form different planar shapes depending on the position along that axis. This concept allows one to calculate the volume of the solid by integrating the area of these slices over the range of values along the axis.

Unit Circle Base

The unit circle, defined by the equation x² + y² = 1, forms the two-dimensional base of the solid. This circle determines the limits of integration along the x-axis and the extent of the solid in the plane, serving as the foundation upon which the vertical cross sections are constructed.

Equilateral Triangle Cross Sections

When the vertical cross sections perpendicular to the x-axis are equilateral triangles, each triangle’s side length is determined by the distance across the circle at that x-value. The area of an equilateral triangle can be computed using the formula (?3/4)*(side length)², and the geometry of the circle is used to express the side length as a function of x.

Integration for Volume Calculation

Once the area of the cross-sectional slice as a function of x is found, calculus is used to compute the volume of the solid by integrating this area function over the appropriate interval. This method, often referred to as the method of slicing, sums up the infinitesimally thin slices to yield the total volume of the solid.

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