A liquid storage tank can be described as the volume generated by revolving the region bounded by the curve y=2016+x2−−−−−−√, and the lines x=0, y=0, and x=2 about the x-axis. Find the volume of the tank (in cubic meters).

V = 25,061.4 m³ or

V= 4032π [2(1 + √126) - √505] m³

Step-by-step explanation:

Given y = 2016 + x²

Since the body is rotated along the y axis, we need to find the limits along y-axis corresponding to x =0 and x=2, we do this by substituting the values of x to obtain the y-values

So at x = 0, y = 4√126

at x = 2, y = 2√505

But V = ∫ πx²dy

So making x the subject we have:

x²= (2016-y)

Therefore V = ∫π(2016-y)dy, at y = 4√126 and 2√505

V = π[2016y- y²]

V = π[[2016(4√126) -(4√126)²-[2016(2√505)-(2√505)²]]

V = 2016π[ [(4√126)(1-4√126)] - [2√505(1-2√505)] ]

V = 2×2016π[ [2√126(1 - 4√126)] - [√505(1 - 2√505)] ]

V= 4032π [ 2√126 - 1008 - √505 +1010]

V= 4032π [2√126 - √505 +2]

V= 4032π [2(1 + √126) - √505]

Or

V = 12672(1.9777)

V= 25061.4 m³