The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x^2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid? May I have all the explanation with the answer, please. Thanks in advantage

Answer 1

Draw the graphs. Each triangle has a base equal to the distance between the graphs: x-x^2
The area of an equilateral triangle of side s is √3/4 s^2
So, add up the volumes of all those thin triangles and you get
v = ∫[0,1] √3/4 (x-x^2)^2 dx = 1/(40√3)