5 · IB Maths
Volume of revolution
Rotate the region under a curve about the x-axis and it sweeps out a solid. Stack thin discs to estimate the volume, then compare with the exact integral.
Predict first
Start from the defaults (y = √x + 0.6, 10 discs, midpoint). If you set 'Disc height taken at the' to left edge, the disc estimate becomes an…
The function is increasing, so the height at the left edge of each disc is smaller than the curve over the rest of the disc. Every disc is too thin, so the total is too small.
Check it in the model: Change the 'Disc height taken at the' menu from midpoint to left edge and read the Error line.
curve and region discs height used for each disc
one disc: ΔV = π y² Δx, Δx = (b − a) ÷ n
partial sweep: V × (angle ÷ 360°)
Try this
- Choose y = 0.4x + 0.6 with a = 0 and b = 5.0. Use the left-edge rule and watch the error as n goes from 5 to 30. Is it an under- or over-estimate, and why?
- Switch to the midpoint rule and double n from 10 to 20. By roughly what factor does the percentage error shrink?
- Set the sweep angle to 180°. Predict the volume from the full revolution, then check both readouts.
Common error: integrating y instead of y². Rotating y = f(x) about the x-axis gives V = π∫y² dx, because each thin disc has area πy²; square the whole function before you integrate.
More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation · Related rates of change · Area under a curve · Calculus of motion · Limits · Maclaurin series · Newton–Raphson method (enrichment) · Optimisation · Slope fields · Tangents and the derivative