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Mathematics
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sub \(u=2x\)
Hmm... but I don't have the function... Im kind of confused

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\[\int\limits_0^2 f(2x)\,dx~~\stackrel{u=2x}{=}~~ \frac{1}{2}\int\limits_0^4 f(u)\,du = \frac{1}{2}*20=10\]
Why does the limit of integration change from 4 to 2?
scratch that, lets work it again from beginning
you want to evaluate \[\int\limits_0^2f(2x)\, dx\] substitute \(u=2x \implies du=2dx\implies dx=\dfrac{du}{2}\)
next work the bounds, as \(x\to 0\), what does \(u\to\) ? as \(x\to 2\), what does \(u\to\) ?
1) x -> 0, u -> 0 2) x -> 2, u -> 4
so upon substitution, bounds change from (0, 2) to (0, 4) and the differential changes from dx to du/2 plug them in
okay, so then that's \[\int\limits_{0}^{4}f(u) \frac{ du }{ 2 }\] and then the 1/2 comes out of the integral
Yes, next recall that the variable in definite integral is "dummy" \[\int\limits_a^b f(\color{red}{x})\,d\color{red}{x} = \int\limits_a^b f(\color{red}{t})\,d\color{red}{t}=\int\limits_a^b f(\color{red}{\spadesuit})\,d\color{red}{\spadesuit}\]
right
Ohhh now I get what's happening!
@ganeshie8 Thanks so much!

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