The base of a solid is the circle x^2 +y^2=9 and each cross section perpendicular to the x-axis is an equilateral triangle with one side across the base. find the volume of the solid.

First, recall that the area of an equilateral triangle with side length is .

Now, each cross section's base is the distance from the top half of the circular base to the lower half. You have



This means the vertical distance is .

So, the volume of such a solid is given by



where the first equality comes from the fact that the circle is symmetric across the y axis.

Answerthe simplified expression isexplanationthe given expression is,let us deal with the bracket first.the two negatives in the parenthesis will become positive.the expression now becomes,we now expand the parenthesis to get,we group like terms to obtain,this simplifies to,