24. Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 - r²) inches per second. (a) In a plane cross section of the pipe, a thin ring with thick- ness Ar at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10-2) (2πr) Ar cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.

2. A cylindrical tank contains oil (use density as p in lb/ ft'). The radius of the tank is 3 feet, the length is 12 feet and oil enters and leaves the tank through an opening at the top. The tank is initially full of oil. Use the sketch below and complete each step below to find the work done in pumping all of the oil out of the opening at the top of the tank. You must use the axis provided. No calculators. (a) Find and expression for the volume of a single representative "slab" of oil that will move out of the tank. (b) Find the expression for the distance any single representative "slab" of oil must move to get out of the tank. (c) Find the expression for the force of a single representative "slab" of oil. (d) Set up (BUT DO NOT SOLVE), the Reimann sum that approximates the total work done in pumping all of the oil out of the tank.