IB Maths AI HLFurther IntegrationPaper 1 & 2~6 min read
Volumes of Revolution
Spin a region around an axis and it sweeps out a 3D solid. Slice that solid into thin discs: each is a circle of radius y (or x) and thickness dx (or dy). Add up the disc volumes with an integral and you get the volume of revolution ā a definite integral with a Ļ out front.
š What you need to know
About the x-axis: V = Ļā«aby2 dx.
About the y-axis: V = Ļā«cdx2 dy.
Square the radius: it’s y2 (or x2) inside the integral, from the disc area Ļr2.
Match axis to variable: rotating about x ā integrate dx with x-limits; about y ā dy with y-limits.
Rearrange for the y-axis: you need x2 as a function of y.
Keep Ļ: leave it in for an exact answer, or multiply out for a decimal.
The disc idea
Volume of revolutionV = Ļā«aby2 dx (about x-axis) ā¢ V = Ļā«cdx2 dy (about y-axis)
ā both forms are in the formula booklet
A region spun into a solid
Each slice is a disc of radius y, area Ļy2, thickness dx. Summing them along the axis gives the volume.
š¤ Why y2 and where does Ļ come from?
Each thin slice is a cylinder (disc) of radius r = y and thickness dx, so its volume is area Ć thickness = Ļy2 dx. The Ļr2 for a circle’s area is exactly where both the Ļ and the square come from. Integrating sums infinitely many such discs.
š§ “Pi, integral, radius squared”
Ļ out front, integral over the limits, the radius squared inside. Rotating about x the radius is y; about y the radius is x. Don’t forget to square it.
Carrying it out
š§ Recipe ā volume of revolution
Identify the axis of rotation ā sets the variable and which radius to square.
Get the radius: y = f(x) for the x-axis; rearrange to x = g(y) for the y-axis.
Square it and multiply by Ļ.
Integrate between the correct limits.
Leave Ļ for an exact answer, or evaluate to a decimal.
Squaring tidies things up: a square root like y = āx becomes y2 = x, and an exponential y = ex becomes y2 = e2x ā both far easier to integrate.
Worked examples
WE 1
The region under y = x from x = 0 to 2 is rotated about the x-axis. Find the volume.
Match the variable to the axis: dx for the x-axis, dy for the y-axis.
Rearrange for x2 when rotating about the y-axis ā and you only need x2, not x.
Keep Ļ for exact answers; multiply out only if asked for a decimal.
Squaring simplifies roots and exponentials ā do it before integrating.
Sanity-check simple solids against cone/sphere formulas.
ā Common mistakes
Forgetting to square the radius ā integrating y instead of y2.
Dropping the Ļ.
Wrong variable ā using dx when rotating about the y-axis.
Squaring wrongly ā e.g. (ex)2 = e2x, not exĀ².
Using x-limits for a rotation about the y-axis.
That wraps up Further Integration. The unit started with the standard integrals and reverse-chain techniques, then turned the definite integral into geometry ā areas under curves, below the axis, against the y-axis, and between graphs ā before lifting it into three dimensions here with volumes of revolution. The disc method is the same definite-integral move once more: sum infinitely many thin slices, this time circles of area Ļr2, between two limits.
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