Calculus I · Unit 3B · lesson
Pumping Liquids
Learning objectives
Build work integrals from slice volume, fluid weight density, and lifting distance.
Pumping Liquids
Every fluid layer travels a different distance
Pumping problems combine three local quantities: the volume of a thin fluid slice, its weight per unit volume, and the distance that slice must be lifted. A horizontal slice at height has volume approximately , weight approximately , and work approximately weight times lifting distance. Integrating over the fluid depth adds the work for all slices.
The lifting distance is often the most commonly omitted factor. It must be measured from the slice's current height to the outlet, not from the bottom of the tank unless those distances happen to coincide. A sketch with the coordinate axis, fluid level, slice, and outlet labeled is essential. The resulting integrand should have units of energy per unit height before multiplication by .
For a horizontal slice at height :
where is weight density, and
Pump water from a rectangular tank
A tank is 4 m long, 3 m wide, and 2 m deep, filled with water. Pump water to an outlet 1 m above the top. Let measure height from the bottom. Cross-sectional area is , lift distance is , and water weight density is approximately N/m:
The setup is the main modeling achievement; evaluation is routine.
u3b-pump-01For the rectangular water tank described in the lesson, write the work integral.
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Use weight density times cross-sectional area times lifting distance.
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A horizontal liquid slice lifted to an outlet. A thin liquid layer at height y has volume A(y)dy and must be lifted D(y). Show slice volume, weight, lift distance, and tank geometry. Keep density, weight density, and mass density distinct.
Every relationship in a horizontal liquid slice lifted to an outlet uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show slice volume, weight, lift distance, and tank geometry.
A horizontal liquid slice lifted to an outlet. Show slice volume, weight, lift distance, and tank geometry.
After the explanation
Use the section idea
Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.
Work adds force through distance, pumping adds slice weight through lift distance, pressure adds depth-dependent strip force, and marginal or probability models add weighted local contributions.
Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.
Confusing mass density with weight density, measuring depth from the wrong reference, or integrating a marginal quantity without an initial value when a total function is requested.
Does the setup respond correctly when the slice moves, and can a units or scale check expose a missing factor?
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