The equilateral triangular cross sections of a solid with base an arch of y = sin(x) on
[0, pi]. find volume analytically

We can define the base of each equilateral section as since this is the distance from the x-axis to the function .

An equilateral triangle's area can have a function defined solely by the base. Since all sides are equal, the height of the triangle can be found via pythagorean theorem as:

But we know that the base is a function of x. We also can then state that Area is also a function of x. Specifically: .

Then if we integrate the area function A(x) from 0 to pi, we get the total volume:

God she she is dudhe es se did she is d