calculus question

Question

set up the integrals that gives the volume of the solid that has as its base the semicircle y=sqrt(4-x^2), where each cross section perpendicular to the x-axis is an equilateral triangle with its base on the circle.Do not integrate

Answer 1

The area of an equilateral triangle depending on its side length s is give by A = s²√3/4.

The semicircle has a radius of 2 so intersects the x-axis at (-2,0) and (2,0). For each x between those two points the segment joining (x, 0) and (x, √(4-x²) is the side of an equilateral triangle (cross section of the solid).

The length of a typical side is then given as √(4 - x²). The area of the equilateral triangle is:

A(x) = √(4 - x²)²√3/4 = (4 - x²)√3/4.

Setting up the definite integral for the volume of a solid with those cross sections can either be:

∫_-2² A(x) dx or 2∫₀² A(x) dx.

Using the latter:

2 ∫₀² [(4 - x²)√3/4}dx = √3/2 ∫₀² (4 - x²)dx

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