Science:Math Exam Resources/Courses/MATH101/April 2011/Question 03 (a)/Solution 1

To solve this problem, we consider moving up the liquid in the cylinder slice by slice.

Consider the slice of the cylinder of height dx as in the picture. Its volume is given by A dx where a is the area of the disc. This is

$\pi (0.5)^{2}\,dx$

Its mass is thus

$1000\pi (0.5)^{2}\,dx=250\pi \,dx$

Since the density of the liquid is 1000 kg/m³. The work to lift the slice to the top is force multiplied by distance which is

$9.8\cdot 250\pi \,dx\cdot x=9.8\cdot 250\pi x\,dx$

where x is the distance from the top of the cylinder to the slice.

Hence the total work is given by integrating this from the top to the bottom of the cylinder, that is from 0 to 2 and we obtain the integral

$W=\int _{0}^{2}9.8\cdot 250\pi x\,dx$