The base of a solid is the circle x2 + y2 = 9. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?
Each triangle has a base 2y. So, since the area of an equilateral triangle of side s is √3/4 s^2, our triangles have area.
√3/4 (9-x^2)
Now, all we have to do is sum up all the triangular plates of thickness dx, from -3<=x<=3. Or, because of symmetry, twice the volume for 0<=x<=3.
v = 2∫[0,3] √3/4 (9-x^2) dx
