Owl: Imagine slicing a loaf of banana bread into coin-thin pieces perpendicular to its length. Each slice has some area A(x). Stack the slices, and the loaf's volume is approximately Σ A(xᵢ) Δx — an integral in disguise.
Disks and washers are special cases: circular slices, sometimes with a hole. The deep idea is always ∫ A(x) dx along the axis you slice perpendicular to.
Why does the area of each thin slice become the integrand — and what goes wrong if you pick the wrong variable of integration?