The inner surface of a bowl is of the shape formed by rotating completely about the y- axis the area bounded by the curve 𝑦 = 𝑥2 − 4 , the x – axis, the y – axis and the line 𝑦 = 3 .
(i) Find the volume of the bowl
(ii) Find the formula that represents the volume of water in the bowl when the depth of water is d (< 3)
If water is poured in at a rate of 5 cubic units per second
(iii) Find the rate at which the depth is increasing when 𝑑 = 1.
(i) using discs of thickness dy, we have
v = ∫[0,3] πr^2 dy
where r = x = y+4
v = ∫[0,3] π(y+4) dy = 33π/2
check, using shells of thickness dx
v1 = 12π
v2 = ∫[2,√7] 2πrrh dx
where r=x and h=3-y
v2 = ∫[2,√7] 2πx(3-(x^2-4)) dx = 9π/2
v = v1+v2 = 33π/2
(ii) v = ∫[0,d] π(y+4) dy = π(1/2 d^2 + 4d)
(iii)
dv/dt = π(y+4) dy/dt
so when d=1,
5 = π(1+4) dy/dt
dy/dt = 1/π