# Separation problem in definite integral of piecewise function

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?

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

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.