# What is the volume of the solid given the base of a solid is the region in the first quadrant bounded by the graph of #y=-x^2+5x-4^ and the x-axis and the cross-sections of the solid perpendicular to the x-axis are equilateral triangles?

The volume is

The curve

For

The solid itself looks something like this:

Now draw the cross-section equilateral triangle, label the sides

By the Pythagorean Theorem,

so that

The cross-sectional area is

Continuing to simplify gives

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The volume of the solid is ( V = \frac{1}{2}\int_{0}^{5} (5x - x^2 - 4)^2 dx ).

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