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**"Work" Related problems**

I'm sure this is a simple answer..Was just curious though why are we multiplying \(\displaystyle \frac{1}{2}\) by everything?

Here is the problem.

if 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14cm, what is the natural length of the spring?

so I said \(\displaystyle \int^{0.12}_{0.10} k*x dx = 6 J\)

which is \(\displaystyle k * \frac{x^2}{2} | 0.10 to 0.12 = 6 J\)

so why are they doing it like this in the book --> \(\displaystyle \frac{1}{2}k * x^2 | 0.10 to 0 .12 = 6j\)

where is the 1/2 randomly coming from?

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Wait are they just multiplying through to get rid of fractions?