What is the volume of the solid whose cross-sections are equilateral triangles perpendicular to the x -axis and with bases on the region bounded by curves y = x 2 + 1 , x = 1 , and x -axis and the y -axis ?

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23-11-2017
Mathematics

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What is the volume of the solid whose cross-sections are equilateral triangles perpendicular to the x -axis and with bases on the region bounded by curves y = x 2 + 1 , x = 1 , and x -axis and the y -axis ?

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wizard123 wizard123 · Answer 1 · 2017-11-23T08:09:24+01:00

The volume is the sum of all the areas of the cross-sections.
[tex]V = \int_a^b A(x) dx[/tex]

The Area is the area of an equilateral triangle, where the length of the base is distance from curve 'x^2 + 1' and x-axis.
The height of an equilateral triangle is
[tex]\frac{\sqrt{3}}{2} b = \frac{\sqrt{3}}{2} (x^2 +1)[/tex]

Therefore Area of triangle is:
[tex]A(x) = \frac{1}{2} b h = \frac{\sqrt{3}}{4} (x^2 + 1)^2[/tex]

Now integrate to find Volume
[tex]V = \frac{\sqrt{3}}{4}\int_0^1 (x^2 + 1)^2 = x^4 +2x^2 +1 \\ \\ =\frac{\sqrt{3}}{4}|_0^1 (\frac{x^5}{5} + \frac{2x^3}{3} + x) \\ \\ =\frac{\sqrt{3}}{4} (\frac{1}{5} + \frac{2}{3} +1) \\ \\ = \frac{7 \sqrt{3}}{15} [/tex]