[Math] A variable force of $5x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin

calculusintegrationphysics

A variable force of $\dfrac{5}{x^2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1$ ft to $x=10$ ft.

I tried doing $\displaystyle \int_1^{10} x\cdot \dfrac{5}{x^2}dx$

then got $\quad \displaystyle 5\int_1^{10} \dfrac{1}{x} dx $

then $5(\ln10-\ln1)$

then $5\ln10$ ft-lb

but the book says $4.5$ ft-lb….

Best Answer

use
$\displaystyle \int_1^{10} \dfrac{5}{x^2}dx$
instead of
$\displaystyle \int_1^{10} x\cdot \dfrac{5}{x^2}dx$

for more info: imagine that $F=5$ then with $\displaystyle \int_1^{10} x\cdot5dx$ formula, the result work will be different from 11 to 20 and from 1 to 10. with this paradox you will find that something is wrong.

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