Question

The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?

Answer 1

the area of a cross section would be \(A(x) = \frac{1}{2} (x-x^2) \times \frac{\sqrt{3}}{2}(x-x^2)\) so \(V =\int_{0}^{1} A(x) \, dx\)