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?

Question

So I got 1/30
because (integral from 0 to 1) (x-x^2)^2=1/30

Answer 1

Not quite. You have figured the volume consisting of squares. For the triangle, the base is x-x^2, so the altitude is (x-x^2)√3/2. The volume is then

∫[0,1] (1/2)(x-x^2)(x-x^2)√3/2 dx
= ∫[0,1] √3/4 (x-x^2)^2 dx = √3/120

Answer 2

for the second integral i got (sqrt 3)/60 how did you get (sqrt 3)/120?

Answer 3

√3/4 (x^5/5 - x^4/2 + x^3/3) [0,1]
= √3/4 (1/5 - 1/2 + 1/3)
= √3/4 * 1/30
= √3/120

Answer 4

Oh thank you!!!