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My question is about separation of $\displaystyle \int _0 ^2 f(x) dx$ defined by $f(x) = \left\{\begin{aligned} &x^2 ,\ x \in [0,1]\\ &x^4+4 ,\ x \in (1,2] \end{aligned} \right.$

Of course, we should write in the following way $\displaystyle \int _0 ^1 f(x) dx + \displaystyle \int _1 ^2 f(x) dx=\displaystyle \int _0 ^1 x^2 dx + \displaystyle \int _1 ^2 x^4+4 \hspace{0.1cm}dx$

I am okay with the expression of first integral. But, we write the second integral as $\int _1 ^2 x^4+4 \hspace{0.1cm}dx$ though at the point $1$ function takes values from $x^2$.

I know this is a standard procedure each bachelor knows. Where is the point I miss?

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    $\begingroup$ Hint: For any integrable function $f: I \subset \mathbb R \to \mathbb R$ and $p \in I$ we have $$\int_I f(x)dx = \int_{I - \{p\}} f(x)dx.$$ $\endgroup$
    – Falcon
    Apr 21 at 20:48

1 Answer 1

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In the second expression, we are adding some extra area which wasn't meant to be added. If you look carefully, the extra area we are referring is the area of a point, (the extra point where $x=1$ and $f(x)=5$). But the thing is, area of a point is $0$ by definition (as it has no dimensions), so it's like we are adding $0$ to the area of second expression.

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