Owl: Imagine a shadow cast by a lamp that stretches farther and farther along a hallway. The hallway never ends — yet the total amount of light spilled on the floor might still be finite if the beam fades fast enough.
That is the puzzle of improper integrals: accumulation when either the interval runs to infinity or the integrand spikes without bound. Infinite stretch does not automatically mean infinite area.
Before we write a single limit, hold two pictures in mind. Under y = 1/x² from x = 1 onward, the curve hugs the axis quickly. Under y = 1/x, the same hallway feels stubbornly thick.
We'll turn both into limits of ordinary integrals and watch which totals settle — and which keep growing forever.