Volume with Disc Method: Revolving Around the x- or y-Axis

Take a flat region between a curve and the x-axis, and spin it all the way around the x-axis. Every vertical strip in that region sweeps out a full circle as it goes around, and since the strip has some tiny width dx, the shape it traces is a thin coin: a disc. Stack up infinitely many of these coins side by side and you get the solid of revolution. That's the whole idea: a solid of revolution is just a pile of discs, and volume is the sum of their volumes.

Each disc is a cylinder with a tiny thickness. Its radius is the distance from the axis to the curve at that point, so if you're spinning around the x-axis, the radius at point x is just y = f(x). The area of that circular face is π times radius squared, so π times f(x) squared. Multiply by the thickness dx and integrate from the left boundary to the right boundary: V = π times the integral of f(x) squared dx. Spinning around the y-axis works the same way, just swap the roles of x and y and integrate with respect to y instead.

This method only works cleanly when the region touches the axis of rotation directly, so every disc is solid with no gap in the middle. The moment the region is lifted off the axis, or you revolve the area between two curves, each slice becomes a washer: a disc with a hole punched out. That's the very next idea, volume stays π times integral of radius squared, but now you subtract the inner radius squared from the outer radius squared before integrating.