Here's the region under y = √x on [0, 4]. Drag the sweep slider to revolve it around the x-axis and watch the solid form. Then flip on "show discs" — the solid is really just a stack of thin discs, and that's the whole secret.
Why does that give a clean formula? Look at one disc. Its radius is the height of the region, R = √x. A disc is just a circle with thickness, so its area is πR² — and stacking all of them is an integral:
The model above is one example. Here's the same machine — but you type the curves, straight from your homework. Pick the axis (x, y, or any line like y=3), hit Revolve, and it hands you the disc / washer / shell setup in your own numbers — then press Sweep to watch the volume build.
Most guides hand you a "disc formula" and a separate "washer formula" and make you memorize which to use. Skip all that. There is one formula for every solid of revolution, and disc is just a special case of it.
R is the outer radius (axis to the far edge of the region); r is the inner radius (axis to the near edge). That's it — every problem is this integral.
So when is it a disc versus a washer? It's the same question as: does the region touch the axis? If it touches, there's no hole — the inner radius is r = 0, and π(R²−0²) collapses to plain πR². A disc is nothing but a washer whose hole has shrunk to nothing.
Stop thinking "which method?" Always write π∫(R²−r²). The only real work — the entire skill — is measuring R and r. That's next.
If one formula covers everything, then every problem comes down to one thing: get R and r right. Three principles handle every case the AP exam can throw at you.
The radius is how far the edge of the region sits from the axis — a distance you measure, not simply "the function." They happen to be equal only when the axis is the x-axis itself (distance from y = 0 is just the height). Move the axis and that shortcut breaks. Always ask "distance from the axis to the edge," and you'll never get tripped up.
The radius always points straight out from the axis — perpendicular to it. That one fact tells you whether to integrate in x or y, which is the step most students guess at:
Horizontal axis → vertical radius → it's made of y's → dx. Vertical axis → horizontal radius → x's → dy (so solve the curves for x first). No guessing.
Every radius is one edge minus another: outer R = far edge − axis, inner r = near edge − axis. When the axis is the x-axis, "− axis" is "− 0" and disappears. When it isn't, you shift by the axis — the step that costs the most points:
Same curve y = √x every time — only the axis moved, so only the "− axis" piece changed. Set it up as far − near and the radius always comes out positive.
Watch the one formula handle all three. Notice each is the same recipe from above — only how you measure the radius changes.
Want to see why that formula is an integral? Step through the whole build on one washer — find its radii, sum the slices, take the limit, land on the integral.
Disc or washer? What's the radius? Where's the axis? Six questions on the decisions that actually get tested.