Visualise solids of revolution in 3D

Direct answer

Revolve curves around the x- or y-axis in 3D and compute the volume with the disc and shell methods.

Spin the yellow curve round an axis to sweep out a solid. Slicing it into discs (about x) or cylindrical shells (about y) turns the volume into an integral.

Simulation

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Solids of Revolution and Volume in 3D

Integration

Curve y = √x

Axis of rotation x-axis

From x = a 0

To x = b 2

Sweep (°) 270°

Drag to rotate, two fingers or right-drag to pan, scroll or pinch to zoom. The toolbar has views, cutaway, measure and “show the maths”.

What's happening? Spin the yellow curve round an axis to sweep out a solid. Slicing it into discs (about x) or cylindrical shells (about y) turns the volume into an integral.

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Where it is in NCERT: Class 12 Mathematics Part II, “Integrals” (Definite Integral, Fundamental Theorem of Calculus, Evaluation of Definite Integrals by Substitution, Some Properties of Definite Integrals) · ncert.nic.in
Topic notes and formulas: Definite Integration
Printable formula sheet
Chapter-wise JEE Main PYQs (14)