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For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
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Is the average force calculated from $F(x)$ the same as that calculated from $F(t)$?
Your reasoning is valid with the important caveat that you yourself realized: this works if $x_1$ corresponds to $t_1$ and $x_2$ corresponds to $t_2$, and as long as the force over time and the force …