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?

1 Answer
Mar 22, 2015

The volume is 813403.5074.

The curve y=x2+5x4=(x1)(x4) has x-intercepts at x=1 and x=4 (and is above the x-axis for 1<x<4).

For 1<x<4, let b(x)=x2+5x4. This will be the base of the equilateral triangle cross-section at x.

The solid itself looks something like this:

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Now draw the cross-section equilateral triangle, label the sides b(x), and draw a vertical line for the height h(x) from the top vertex down perpendicular to the base.
By the Pythagorean Theorem, (b(x))2=(b(x)2)2+(h(x))2.

enter image source here

so that h(x)=(b(x))2(b(x)2)2=34(b(x))2=32b(x).

The cross-sectional area is A(x)=12b(x)h(x)=34(b(x))2=34(x410x3+33x240x+16), which means the volume of the solid is V=41A(x)dx=3441(x410x3+33x240x+16)dx=34(x555x42+11x320x2+16x)x=4x=1.

Continuing to simplify gives V=34((1024512802+704320+64)(1552+1120+16))=34(6454710)=81340.