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 volume of the solid can be calculated using the formula V = (1/3)Ah, where A is the area of the base and h is the height of the solid.

The area of the base is the area of the first-quadrant region bounded by y = x and y = x2. This can be calculated using the formula A = ∫x2 - x dx, which gives A = (1/3)x3 - (1/2)x2 + c.

The height of the solid is the length of the side of the equilateral triangle, which is equal to 2√3.

Therefore, the volume of the solid is V = (1/3)(1/3)x3 - (1/2)x2 + c)(2√3) = (2/9)x3 - (1/3)x2 + c√3.