Maths (university) · Part 05
Integration
Areas, antiderivatives and the link between integrals and derivatives.
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- AntiderivativeA function F whose derivative is f. Because constants vanish when you differentiate, every antiderivative comes with + C.
- Power rule for integralsThe integral of x^n dx is x^(n+1) / (n+1) + C, for n not equal to -1. Raise the power by one, then divide by the new power.
- Riemann sumAn estimate of area under a curve made by adding up thin rectangles. Using more rectangles gives a better answer, and the limit is the definite integral.
- Definite integralThe integral of f from a to b is the signed area between the curve and the x-axis. Areas above the axis count positive, below count negative.
- Fundamental theorem of calculusIf F' = f, then the integral of f from a to b equals F(b) - F(a). Also, the derivative of the integral from a to x of f(t) dt is f(x).
- Standard integralsReverse the standard derivatives: cos x integrates to sin x, sin x to -cos x, e^x to e^x and 1/x to ln|x|, each plus C.
- Integration by substitutionReplace a tangled part with a new letter u so the integral becomes simple. Change dx into du too, and change limits if the integral is definite.
- Area between curvesIntegrate the top curve minus the bottom curve over the interval where they enclose a region. Find the crossing points first.
- Average valueThe average value of f on [a, b] is the integral of f over [a, b], divided by the length b - a.
- Signed area and net changeAn integral adds area above the axis and subtracts area below it. So it gives net change, which can cancel out to zero even when the true area does not.