The base of a solid is a circle of radius 4. Find the volume of the solid if each cross-section perpendicular to a fixed diameter of the circle is an isosceles triangle with height equal to its base.
Volume of an object with given cross-section equation
The Attempt at a Solution
I know how to solve this (and similar questions) without problem, I just want to know if there is another way to solve them than the one I am using. For this question I found the equation of the circle (which is y=sqrt(16-x^2)) and then the area formula for the isosceles triangle (which I found to be y=b^2/2). Using the cross-section equation, I set the problem up as dV=(1/2)b^2dx, and then using the equation of the circle in place of b: dV=(8-(x^2/2))dx. To finally solve the question I then integrated this formula over [-4,4] and came up with an answer of 128/3, which I then multiplied by 4 to find the correct answer of 512/3. What I want to know is the 'proper' way to solve this question rather than having to multiply by 4 at the end. I don't know if all questions will allow me to multiply by 4 (or 1/4) at the end to find the correct answer, so i'm hoping someone can show me another way to go about this question. If anyone can help it would be greatly appreciated, thanks in advance.