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Pumping Gasoline In Exercises, find the work done in pumping gasoline that weighs 42 pounds per cubic foot. (Hint: Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.) A cylindrical gasoline tank 3 feet in diameter and 4 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire contents of the fuel tank into a tractor.

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ f(y)=y^{2}+1, g(y)=0, \quad y=-1, \quad y=2 $$

Lifting a Chain, consider a 15-foot chain that weighs 3 pounds per foot hanging from a winch 15 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Wind up one-third of the chain.

Lifting a Chain, consider a 15-foot chain that weighs 3 pounds per foot hanging from a winch 15 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Run the winch until the bottom of the chain is at the 10 -foot level.

Length of Pursuit \(\mathrm{A}\) fleeing object leaves the origin and moves up the \(y\) -axis (see figure). At the same time, a pursuer leaves the point \((1,0)\) and always moves toward the fleeing object. The pursuer's speed is twice that of the fleeing object. The equation of the path is modeled by \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) How far has the fleeing object traveled when it is caught? Show that the pursuer has traveled twice as far.

Demolition Crane, consider a demolition crane with a 500 -pound ball suspended from a 40 -foot cable that weighs 1 pound per foot. Find the work required to wind up 15 feet of the apparatus.

Find the arc length from \((-3,4)\) clockwise to \((4,3)\) along the circle \(x^{2}+y^{2}=25 .\) Show that the result is one-fourth the circumference of the circle.

Boyle's Law , find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume. (See Example 6.) A quantity of gas with an initial volume of 2 cubic feet and a pressure of 1000 pounds per square foot expands to a volume of 3 cubic feet.

A solid is generated by revolving the region bounded by \(y=\frac{1}{2} x^{2}\) and \(y=2\) about the \(y\) -axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-fourth of the volume is removed. Find the diameter of the hole.

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis.\(y=\frac{x^{3}}{6}+\frac{1}{2 x}, \quad 1 \leq x \leq 2\)