Mathematics M2 Disc method

y = x^2 - 5x + 4 = (x - 5/2)^2 - 25/4 + 4 = (x - 5/2)^2 - 9/4
so x = +/- sqrt (y + 9/4) + 5/2
so the curve can be split into 2 parts : x1 = sqrt ( y + 9/4) + 5/2
and x2 = - sqrt(y + 9/4) + 5/2.
and vertex is (5/2, - 9/4)
so volume of revolution = ∫ πx1^2 dy - ∫ πx2^2 dy from y = - 9/4 to 0
= π ∫ (x1 - x2)(x1 + x2) dy
= π ∫ 5[2 sqrt (y + 9/4] dy
= 10π ∫ sqrt ( y + 9/4) dy from y = - 9/4 to 0
= 20π/3 ( y + 9/4)^(3/2) from y = - 9/4 to 0
= (20π/3)(9/4)^3/2
= (20π/3)(27/8) = 45π/2.