Science:Math Exam Resources/Courses/MATH103/April 2016/Question 05 (b)/Solution 1

The density function of the liquid is given in the question as $\rho (x,y,z)=e^{-y}.$

By the hint, the mass $m_{S}$ of the solid is

$m_{S}=\int _{S}e^{-y}\,dV.$

We consider slices of the solid $S$ made at heights $y$ above the x-axis, where each slice has thickness $dy$ and cross-sectional area $A(y).$

In the case of our solid, the cross-sections at height $y$ are circles with radii $e^{-y}$, so that $A(y)=\pi e^{-2y}.$

Therefore $dV=A(y)\,dy=\pi e^{-2y}\,dy$ (for $0\leq y\leq 1$).

Returning to the mass formula, this implies that

${\begin{aligned}m_{S}&=\int _{S}e^{-y}\,dV\\&=\int _{0}^{1}e^{-y}\cdot \pi e^{-2y}\,dy\\&=\pi \int _{0}^{1}e^{-3y}\,dy\\&=-{\frac {\pi }{3}}\cdot \left.e^{-3y}\right|_{y=0}^{y=1}\\&={\frac {\pi }{3}}(1-e^{-3}).\end{aligned}}$

Hence the mass of the liquid inside the solid is $\color {blue}{\frac {\pi }{3}}(1-e^{-3})\,\mathrm {g} .$