Calculus Help with volume of a solid
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Calculus Help with volume of a solid

[From: ] [author: ] [Date: 12-04-13] [Hit: ]
im not even sure where to start!Thanks-So first, draw the domain.Now, we were given that the cross sections perpendicular to the x-axis will be equilateral triangles, so we need the length of the base.......
Let the base of a solid be the first quadrant plane region bounded by y=1-x^2/25 , the x-axis, and the y-axis. Suppose that cross sections perpendicular to the x-axis are equilateral triangles sitting on the base. Find the volume of the solid.

I'm completely lost on this, i'm not even sure where to start!
Thanks

-
So first, draw the domain. (see link)

Now, we were given that the cross sections perpendicular to the x-axis will be equilateral triangles, so we need the length of the base. This will be the difference between the function and the x-axis, or just y = f(x).

If we do a little algebra on the triangle area formula,
A = ½ b h
Equilateral triangles can be broken into 2 triangles of (30-60-90)° angles, which have proportions (s/2, s, s√(3)/2):
A = [2 * 1/2] * [(s/2) * (s√(3)/2)]
A = s²√(3)/4

In this case, our side length was already determined to be y = f(x), so we can just plug that in:
A = y²√(3)/4
Then we need to integrate over the x-axis from 0 to 5.
V = ∫ A(x) dx
V = √(3)/4 ∫ (1 - x²/25)² dx[0,5]
And I hope you can take it from there. Just expand the square and integrate term-by-term with the power rule.
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