Q 63. - Calculus

Question

Emmy oversees the operation of a sedimentation lagoon that was built and lined using the natural contours of the terrain. The bottom of the lagoon is the part of the surface

$z=\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3$

that lies below the $z=0$plane, where all units are in meters and $z=0$

represents the water level. What is the volume of the lagoon?

Step-by-Step Solution

Answer

The volume of lagoon $1375$ cubic meters.

Step 1: Given Information

It is given that $z=\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3$, $z=0$.

Step 2: Graph

The graph is as below:

Step 3: Simplification

The range is below:

$-15\le x\le 15,-15\le y\le 15$

Hence, volume

${\iint }_{\Omega }\left(\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3\right)dxdy={\int }_{-15}^{15}{\int }_{-15}^{15}\left(\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3\right)dxdy$

${\int }_{-15}^{15}{\int }_{-15}^{15}\left(\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3\right)dxdy=-1375$

$\left|{\int }_{-15}^{15}{\int }_{-15}^{15}\left(\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3\right)dxdy\right|=1375$

Volume of lagoon is $1375$ cubic meters.