Question: The problem of finding the work done in lifting a payload from the surface of the moon is another type of work problem. Suppose the moon has a radius of \(R\) miles and the payload weighs \(P\) pounds at the surface of the moon (at a distance of \(R\) miles from the surface of the moon). When the payload is \(x\) miles from the center of the moon \((x \geq R)\), the gravitational attraction between the moon and the payload is given by the following relation:

\[\text { required force }=f(x)=\frac{R^{2} P}{x^{2}} \text { . }\]

1. The total amount of work done raising the payload from the surface (x = R) to an altitude of R(x=R+R=2 R) is
   \[\text{ work }=\int_{a}^{b}{f}(x)dx=\int_{R}^{2R}{\frac{{{R}^{2}}P}{{{x}^{2}}}}dx=s\]
2. How much work will be needed to raise the payload from the altitude R above the surface (i.e., x = 2 R to an altitude of 2R?
3. How much work will be needed to raise the payload from the surface to an altitude of 2R?

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