Science:Math Exam Resources/Courses/MATH101/April 2018/Question 07 (ii)/Statement

Let ${\textstyle {\mathcal {R}}}$ be the portion of the circle ${\textstyle (y+1)^{2}+x^{2}=4}$, with distances measured in metres, that lies to the right of the ${\textstyle y}$-axis and is below the horizontal line ${\textstyle y=0}$ denoting the ${\textstyle x}$-axis (see the plot below). A reservoir dug into the ground has the shape that results from the region ${\textstyle {\mathcal {R}}}$ being rotated around the ${\textstyle y}$-axis. A cross-section of the reservoir is shown below.

(ii) The reservoir from part (i) is completely filled with a fluid of density ${\textstyle 1000\,{{\mbox{kg}}/{\mbox{m}}^{3}}}$. Find the work, in joules, required to pump all the fluid out of the top of the reservoir. Use ${\textstyle g=9.8}$ m${\textstyle {}/{}}$s${\textstyle {}^{2}}$ for the acceleration due to gravity. A calculator-ready answer is sufficient.