[Math] Integration by substitution, why do we change the limits

Question

I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make?

Source of Quotation: Calculus: Early Transcendentals, 7th Edition, James Stewart

Answer 1

Because the function has changed. Let's do an example:

$$\int_{-1}^1 x\,dx =0$$

because the integrand is odd and the interval is symmetric (you can also check directly).

Let's do a simple $u=x+1$ so that $du=dx$ then the right way we have:

$$\int_0^2 u-1\,du = {1\over 2}(u-1)^2\bigg|_0^2={1\over 2}-{1\over 2}=0$$

but if we fail to change the limits:

$$\int_{-1}^1(u-1)\,du = {1\over 2}(u-1)^2\bigg|_{-1}^1=-2$$

The underlying reason is that integration comes from Riemann sums, the function values depend on the interval of integration. When you change the interval, the heights of the rectangles you use in the definition change (remember the heights are the function values) so that you end up adding up different things if you don't change the function to compensate.