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Unit 6
15–20% of examThis unit turns the derivative around. If you know how fast something is changing, the integral tells you the net change, which you can picture as the area between the rate graph and the axis. You'll estimate that area with Riemann sums, connect it to antiderivatives through the Fundamental Theorem of Calculus, and learn techniques for finding antiderivatives. BC students add integration by parts, partial fractions and improper integrals.
Longer videos that cover the whole unit. Good for a first pass or a final review.
If you graph a rate (like gallons per minute) against time, the area between the graph and the time axis is the total amount that built up. Area below the axis counts as negative change, and the units of the area are the rate's units times the time units.
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You can estimate the area under a curve by adding up rectangles (left, right or midpoint Riemann sums) or trapezoids, even from a table whose intervals aren't equal widths. Whether the estimate is too high or too low depends on whether the function is increasing or decreasing (for left and right sums) or on its concavity (for midpoint and trapezoidal sums).
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A Riemann sum can be written compactly with sigma (Σ) notation. As the number of rectangles goes to infinity and their widths shrink to 0, the limit of the sum is the definite integral ∫ from a to b of f(x) dx, and you should be able to translate between the two forms.
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An accumulation function g(x) = ∫ from a to x of f(t) dt measures area built up from a to x. The Fundamental Theorem of Calculus says that if f is continuous, g′(x) = f(x), so the derivative of an accumulation function is just the function inside. If the upper limit is something like x², you multiply by its derivative too (chain rule).
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Because g′ = f, you can read the behavior of g(x) = ∫ from a to x of f(t) dt straight from the graph of f: g increases where f is positive, has a max or min where f changes sign, and is concave up where f is increasing. You find actual values of g by adding and subtracting areas.
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Definite integrals follow simple rules. Swapping the limits flips the sign, an integral from a to a is 0, you can split an interval into pieces and add them, and constants and sums can be pulled apart. You can also find some definite integrals exactly using geometry, like areas of triangles, rectangles and semicircles.
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To evaluate ∫ from a to b of f(x) dx exactly, find any antiderivative F (where F′ = f) and compute F(b) − F(a). This also means the total change in a quantity equals the integral of its rate of change.
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An antiderivative of f is a function whose derivative is f, and the indefinite integral ∫ f(x) dx = F(x) + C stands for all of them. Learn the basic rules by reversing derivative rules: the power rule (when the power isn't −1), ∫ (1/x) dx = ln|x| + C, and the integrals of eˣ, sin x, cos x and sec² x.
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u-substitution reverses the chain rule: pick an inner expression u whose derivative also shows up in the integrand, rewrite everything in terms of u, and integrate. For definite integrals, either change the limits to u-values or substitute back to x before plugging in.
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Some rational functions need algebra before you can integrate them. If the top's degree is at least the bottom's, use long division first. If the bottom is a quadratic with no real roots, completing the square can turn it into a form that integrates to an arctangent, since ∫ 1/(1 + u²) du = arctan u + C.
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Integration by parts reverses the product rule: ∫ u dv = uv − ∫ v du. It works well for integrals like ∫ x·eˣ dx, ∫ x·sin x dx or ∫ ln x dx. Choose u to be the part that gets simpler when you differentiate it, so the new integral is easier.
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When a rational function's denominator factors into different (non-repeating) linear factors, you can split it into simpler fractions like A/(x − a) + B/(x − b). Each piece then integrates to a natural log.
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An improper integral has an infinite limit of integration or an integrand that blows up (is unbounded) somewhere on the interval. You handle it by replacing the problem spot with a variable, integrating, and taking a limit: if the limit is a finite number the integral converges, otherwise it diverges.
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This topic is about deciding which method fits an integral: a basic rule, rewriting with algebra or a trig identity, u-substitution, long division or completing the square (plus integration by parts and partial fractions in BC). Practice spotting the pattern before you start calculating.
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