In my test I used limits when I calculated the integral of a piecewise function. In my head the reasoning was that it was an improper integral in one piece of the function because it never reached that point although the limit is easily solved with simple substitution.
$f(x)=\begin{cases}x+3 & x\in\mathbb{[-2;1]}\\ (x-2)^2 & x\in\mathbb{(1;3]} \\ \end{cases}$
This was the function, so what I did is to calculate the integral of the function I divided the function in two different named function lets say $ g(x)=x+3 $ and $ h(x)=(x-2)^2 $. As we can see both functions have limited domains for g(x) the domain is $[-2;1]$ and for h(x) the domain is $(1;3]$. So what I did to calculate the integral of f(x) was:
$ \int_{-2}^3 f(x) = \int_{-2}^1g(x) + \lim_{a \to 1^+}\int_a^3h(x)$
I did this because $h(x)$ is never really $1$, so when calculating the integral of a function that does not contain the bound because it never really reaches the bound I use improper integrals, and that is why I decided to use limit. To solve it I just used simple substitution.
I cant understand why that would be wrong, my math teacher took points of my test score because of this but he wasn't able to explain why this was wrong, and frankly I can't understand what is wrong either. So my question is: is this really wrong? And if it is, why?
Thanks :)