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Integrals
An integral adds up infinitely many infinitely thin pieces. Most often, that adds up to an area under a curve.
Riemann sum
∫₀⁴ (½x + ½) dx · n = 8
slices8
Sum of slice areas
5.500
True area
6
More slices → closer to the true value.
The big idea
To find the area under a curve between two x-values, slice it into tall thin rectangles, find each rectangle's area, and add them up. The thinner the slices, the closer your sum gets to the true area. The integral is what you get in the limit.
Notation
We write ∫ab f(x) dx. Read it as: "sum, from a to b, of f(x) times tiny dx."
The Fundamental Theorem of Calculus
Differentiation and integration are inverses. To compute
∫ab f(x) dx, find any function F whose derivative is f, and compute F(b) − F(a). The messy infinite sum becomes a one-line subtraction. This is the single most important result in calculus.Example
To integrate ½x + ½ from 0 to 4: an antiderivative is F(x) = ¼x² + ½x. So the area is F(4) − F(0) = (4 + 2) − 0 = 6.