9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)


The work required to empty the top half of the tank

Verified step by step guidance

1. Identify the portion of the tank to be emptied: since the tank height is 10 m, the top half corresponds to the water from height 5 m to 10 m.
2. Set up the integral limits accordingly: the variable of integration \( y \) will go from 5 to 10, representing the vertical position of the water slice being pumped.
3. Recall the integrand from the original problem: \( 25 \pi \rho g (15 - y) \), where \( 25 \pi \) is the cross-sectional area (since radius \( r = 5 \), area \( = \pi r^2 = 25 \pi \)), \( \rho \) is the density of water, \( g \) is acceleration due to gravity, and \( (15 - y) \) is the distance the water must be lifted to the outflow pipe 15 m above the bottom.
4. Write the revised integral for the work to pump out the top half of the tank as \( \int_{5}^{10} 25 \pi \rho g (15 - y) \, dy \).
5. This integral represents the total work required to pump all the water from height 5 m up to 10 m out through the pipe at 15 m.

Key Concepts

Work as an Integral in Fluid Mechanics

Work done to pump fluid is calculated by integrating the force needed to move each layer of fluid times the distance it must be moved. This involves setting up an integral where the integrand represents the weight of a thin slice of water multiplied by the vertical distance to the outflow point.

Setting Limits of Integration for Partial Volumes

When calculating work for only part of the tank, such as the top half, the limits of integration must correspond to the vertical range of that portion. For the top half of a 10 m tank, the integral limits change from 0 to 10 to 5 to 10, reflecting the height interval of the water being pumped.

Distance Function in Work Integrals

The distance each water slice is pumped depends on its vertical position. If the outflow pipe is 15 m above the bottom, the distance function is (15 - y), where y is the height of the slice. This distance varies with y and must be included in the integrand to correctly calculate work.