definite integration

a) Equation of OA is y = x

Equation of OB, by point-slope form, is:

(y - 1)/(x - 1) = (0 - 1)/(3 - 1)

2y - 2 = 1 - x

x + 2y - 3 = 0

b) i) When the whole region bounded by the axes, x = 3 and y = 1 is revolved about the x-axis, the volume of the cylinder is 3π cubic units.

So when y = x is revolved about the x-axis from y = 0 to 1, the outer vol. (NOT the red region) is:

2π ∫ (y = 0 → 1) xy dy

= 2π ∫ (y = 0 → 1) y2 dy

= 2π/3 cubic unts

When y = 3 - 2x is revolved about the x-axis from y = 0 to 1, the outer vol. (NOT the red region) is:

2π ∫ (y = 0 → 1) xy dy

= 2π ∫ (y = 0 → 1) y(3 - 2y) dy

= 2π [3y2/2 - 2y3/3] (y = 0 → 1)

= 5π/3 cubic units

So the required vol. is 3π - 2π/3 - 5π/3 = 2π/3 cubic units.

ii) 2π ∫ (x = 0 → 1) x(x) dx + π ∫ (x = 1 → 3) x(3 - x) dx

= 2π [x3/3] (x = 0 → 1) + π [3x2/2 - x3/3] (x = 1 → 3)

= 2π/3 + 13π/6

= 17π/6 cubic units.

iii) 2π ∫ (x = 0 → 1) (3 - x)(x) dx + π ∫ (x = 1 → 3) (x - 3)(3 - x) dx

= 2π [3x2/2 - x3/3] (x = 0 → 1) - π [(x - 3)2/2] (x = 1 → 3)

= 7π/3 + 2π

= 13π/3 cubic units