Study Cases
Application of Integrals~ Week 13
1 Case Study 1
Area Between Curves – Open Pit Mine Cross-Section: A mining engineer analyzes the cross-sectional area of an open pit mine. The pit walls are modeled by two functions:
\[ \begin{eqnarray*} y_1(x) &=& 0.03x^2 + 4 \\ y_2(x) &=& 0.6x + 1 \end{eqnarray*} \]
where \(x\) and \(y\) are measured in meters.
Tasks
- Determine the integration limits.
- Compute the area of the pit cross-section.
2 Case Study 2
Ore Body Area Above Cut-Off Grade: An ore body profile above the cut-off boundary is approximated by:
\[ y = 25 - 0.02x^2 \]
The cut-off grade boundary is given by:
\[ y = 7 \]
Tasks
- Compute the area of the mineable ore body.
- Adjust the area using a recovery factor of 85%.
3 Case Study 3
Drainage Area of Underground Mine Tunnel: The drainage cross-section of an underground tunnel is modeled by:
\[ y = 10 - x^2 \]
Tasks
- Compute the cross-sectional drainage area.
- Estimate the water inflow capacity if each \(1,m^2\) can drain 1.5 m\(^3\)/hour.
4 Case Study 4
Volume of Overburden – Disk Method: The vertical profile of an overburden layer in a surface mine is given by:
\[ y = 15 - 0.01x^2 \]
The region is rotated about the \(x\)-axis.
Tasks
- Identify the appropriate volume method.
- Compute the overburden volume.
5 Case Study 5
Volume of Mine Shaft – Solid of Revolution: The wall profile of a mine shaft is modeled by:
\[ y = 0.03x^2 \]
The curve is rotated about the \(y\)-axis from \(y = 0\) to \(y = 12\) meters.
Tasks
- Compute the shaft volume.
- Determine the usable volume if structural supports occupy 10% of the space.
6 Case Study 6
Ore Layer Volume – Washer Method: An ore layer is bounded by:
\[ \begin{eqnarray*} y_1(x) &=& 22 - 0.015x^2 \\ y_2(x) &=& 6 \end{eqnarray*} \]
The region is rotated about the \(x\)-axis.
Tasks
- Identify the inner and outer radii.
- Compute the volume of extractable ore.
7 Case Study 7
Stockpile Volume – Shell Method: The profile of a mineral stockpile is modeled by:
\[ x = 0.4y^2 \]
The region is rotated about the \(y\)-axis from \(y = 0\) to \(y = 9\) meters.
Tasks
- Identify the shell radius and height.
- Compute the stockpile volume using the shell method.
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