6Unit 6: Integration and Accumulation of Change

If you know the RATE at which something changes, adding up (accumulating) that rate over time gives the TOTAL change.

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To estimate the area under a curve, slice the region into rectangles and add their areas.

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As the rectangles get infinitely thin, the Riemann sum becomes exact — that limit is the definite integral.

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An accumulation function $g(x)=\int_a^x f(t)\,dt$ builds up area as $x$ moves right.

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Because $g'=f$, the shape of an accumulation function $g$ is read from $f$: $g$ rises where $f>0$, has extrema where $f$ changes sign, and is concave up where $f$ is increasing.

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Definite integrals behave like sums: they split over adjoining intervals, pull out constants, add over sums, and flip sign when you reverse the limits.

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To get the exact area, find an antiderivative and subtract its values at the two limits — no rectangles needed.

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Antidifferentiation reverses the derivative.

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Substitution reverses the chain rule: spot an inner function and its derivative, rename the inner function $u$, and the integral simplifies.

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When a rational integrand is “top-heavy” or leads to an inverse-trig form, reshape it first: long-divide improper fractions, or complete the square in a denominator.

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For a product of two different kinds of function, trade the integral for an easier one using the reverse of the product rule.

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A rational function with a factorable denominator can be split into simpler fractions, each easy to integrate as a logarithm.

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When a limit of integration is infinite (or the integrand blows up), define the integral as a LIMIT and see whether it converges to a finite value.

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Read the form of the integrand first, then pick the tool: basic rule, u-substitution, or (BC) parts / partial fractions / a reshaping step.

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