Yes, you're correct. If the function changes at a certain point along the path of integration (such as a piecewise function or if you're integrating over regions where different formulas apply), then the integral should indeed be split up into separate intervals. This ensures that each part of the function is integrated over the correct region where it applies.

For instance, if you are integrating a function $f(x)$ over an interval $[a, b]$, and $f(x)$ changes form at some point $c$ where $a < c < b$, then the integral should be split into:

$\int_a^b f(x)\, dx = \int_a^c f_1(x)\, dx + \int_c^b f_2(x)\, dx$

Where $f_1(x)$ applies to the interval $[a, c]$ and $f_2(x)$ applies to $[c, b]$.

Would you like further clarification on this concept or how to handle such integrals?

Here are 5 related questions to expand on the topic:

1. How do you handle integration if there are multiple points where the function changes?
2. What is the significance of continuity when splitting integrals?
3. Can you provide an example of a piecewise function and demonstrate how to split the integral?
4. How does the Fundamental Theorem of Calculus apply when splitting integrals?
5. What is the best way to handle discontinuities during integration?

Tip: When dealing with piecewise functions, always check for continuity at the points where the function changes, as it may affect the integral's value.