The limits of integration are really important when we want to calculate definite integrals. They help us figure out the result by looking at how the function behaves between the two points we choose.

When we calculate a definite integral, we use two limits, let's call them $a$ and $b$. These limits set the range where we are finding the area under the curve of the function $f(x)$. We can write this as:

$$\int_a^b f(x) \, dx$$

If we switch the limits around, like changing $a$ and $b$ to $b$ and $a$, the value of the integral changes its sign. This means:

$$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$$

Also, if one of the limits goes to infinity, for example, if $a$ is a number and $b$ becomes infinity, we look at how the function behaves when there is no end:

$$\int_a^\infty f(x) \, dx$$

Doing this helps us understand if the integral will keep growing or if it settles down to a certain value.

It's also important to think about whether the function $f(x)$ is continuous and bounded between the chosen limits. This can seriously change what we find as the answer.

On the other hand, there are indefinite integrals. These don't have the same limits and instead show a whole set of functions. They look like this:

$$\int f(x) \, dx = F(x) + C$$

Here, $F(x)$ is called the antiderivative of $f(x)$, and $C$ is a constant.

So, while definite integrals give us specific numbers based on the limits we pick, indefinite integrals show us a general form that can represent many functions.