Study Cases
Application of Integrals~ Week 13
1 Case Study 1
Area Between Curves – Furnace Cross-Section:
A metallurgical engineer wants to estimate the vertical cross-sectional area of an industrial melting furnace bounded by the inner and outer furnace walls. The profiles of the two walls are modeled as:
\[ \begin{eqnarray*} y_1(x) &=& 0.02x^2 + 5 \\ y_2(x) &=& 0.4x + 2 \end{eqnarray*} \]
where \(x\) and \(y\) are measured in meters.
Tasks
- Identify the limits of integration for the bounded region.
- Compute the cross-sectional area of the furnace using definite integrals.
2 Case Study 2
Area of Molten Metal Above the Slag Line:
The surface profile of molten metal inside a melting furnace is modeled by:
\[ y = 20 - 0.01x^2 \]
The slag–metal interface (slag line) is located at:
\[ y = 5 \]
Tasks
- Compute the cross-sectional area of the molten metal zone.
- Adjust the area using an effective metal yield of 22%.
3 Case Study 3
Flow Area of Molten Metal in a Gating System:
The cross-sectional profile of molten metal flow in a casting runner is approximated by:
\[ y = 12 - x^2 \]
Tasks
- Determine the flow cross-sectional area using integration.
- Estimate the initial molten metal quantity if each \(1\,m^2\) of area corresponds to 0.8 metric tons of metal.
4 Case Study 4
Crucible Volume – Disk Method:
The vertical cross-section of a crucible used for metal melting is given by:
\[ y = 10 - 0.005x^2 \]
The region is rotated about the \(x\)-axis.
Tasks
- Identify the appropriate volume calculation method.
- Compute the volume of the crucible.
5 Case Study 5
Volume of a Metal Casting Mold:
The profile of a metal casting mold cavity is described by:
\[ y = 0.02x^2 \]
The curve is rotated about the \(y\)-axis from \(y = 0\) to \(y = 8\) meters.
Tasks
- Compute the mold volume.
- Determine the effective molten metal volume if the mold is filled to 85% capacity.
6 Case Study 6
Volume of a Molten Metal Layer – Washer Method:
A layer of molten metal inside a furnace is bounded by:
\[ \begin{eqnarray*} y_1(x) &=& 18 - 0.02x^2 \\ y_2(x) &=& 5 \end{eqnarray*} \]
The region is rotated about the \(x\)-axis.
Tasks
- Identify the outer and inner radii for the washer method.
- Compute the volume of the molten metal layer.
7 Case Study 7
Volume of a Metallurgical Reactor Vessel – Shell Method:
The inner wall geometry of a metallurgical reactor vessel is defined by:
\[ x = 0.5y^2 \]
The region is rotated about the \(y\)-axis from \(y = 0\) to \(y = 10\) meters.
Tasks
- Identify the shell radius and shell height.
- Compute the reactor vessel volume using the shell method.
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