Stack of thin discs: Spin the region under y = f(x) around the x-axis. Slice the solid into thin discs: each disc has radius y and thickness dx, so its area is πy². Adding all the discs gives the volume.
V = ∫ π y² dx — square y first, then integrate.
IB-style question — a conical lampshade
The line y = x + 1 between x = 0 and x = 3 is rotated 360° about the x-axis to model a lampshade (cm).
Find the volume of the solid formed.
Step by step
- Use the x-axis formula with y = x + 1.
- Integrate (reverse chain on (x+1)²).
- Substitute the limits.
- Evaluate (or use the GDC).
Final answer
The lampshade's volume is 21π ≈ 66.0 cm³.
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Now the radius is x: Spin the region around the y-axis instead. Now each disc has radius x and thickness dy, so V = ∫ π x² dy. Rewrite x² in terms of y and integrate between the y-values.
This is the AI HL favourite — bowls, vases and glasses are y-axis solids.
IB-style question — a bowl
The curve y = x² for 0 ≤ x ≤ 2 is rotated 360° about the y-axis to make a bowl (cm).
Find the volume of the bowl.
Step by step
- About the y-axis we need x² in terms of y. From y = x², x² = y.
- Find the y-limits: x runs 0 → 2, so y = x² runs 0 → 4.
- Apply V = ∫ π x² dy.
- Evaluate.
Final answer
The bowl's volume is 8π ≈ 25.1 cm³. (Key step: convert to x² in terms of y and use the y-limits.)