Area Under a Curve and Solids of Revolution

Also called

area under a curve · Riemann sum · definite integral · integration as a limit of a sum · rectangles under a curve · lower sum · area under sin x · solid of revolution · volume of revolution · disc method · π∫y²dx · fundamental theorem of calculus · समाकलन · निश्चित समाकल · वक्र के नीचे का क्षेत्रफल

Topics

• The area under y = f(x) from a to b is the limit of the sum of thin rectangles, Σ f(x) Δx, as Δx → 0.
• With more, thinner rectangles the sum creeps up and settles on one number: the definite integral ∫ f(x) dx.
• Under one arch of sin x the rectangles give 1.43, 1.73, 1.87, 1.93 … approaching 2.
• The exact area is ∫₀^π sin x dx = [−cos x] from 0 to π = 1 + 1 = 2 square units.
• Spinning the region under the curve once about the x-axis sweeps out a solid of revolution.
• Each cross-section of that solid at x is a circle of radius y, so a thin slice is a disc of volume πy² dx.
• Adding thinner and thinner discs gives V = π∫ y² dx; for one arch of sin x that is π²/2 ≈ 4.93.
• These rectangles and discs sit inside the shape, so every sum falls a little short of the exact value.

About this model

A GLB file of about 808 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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