Application of Integrals

Question

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?

Answer 1

The area in the xy plane is between x=0 and x=1

The base of the triangle involved is (x-x^2).The altitude of the equilateral triangle is (3^.5)(x-x^2)/2.

The area of the equilateral triangle is .25*(3^.5)*(x-x^2)(x-x^2).

The volume is .25*(3^.5) * integral from 0 to 1 of (x^2-2x^3+x^4).

The integral to be evaluated between 0 and 1 is (x^3)/3-2(x^4)/4+(x^5)/5 which evaluates to 1/30.

The end result is (3^.5)/120=.014433.