Chop the interval, sample a height on each piece, sum the rectangles. Every method is a choice of where to sample.
Over or under?
\(f\) increasingLeft under · Right over (decreasing flips both)
\(f\) concave upTrapezoid over · Midpoint under (concave down flips both)
error scalingL/R \(\sim 1/n\); M/T \(\sim 1/n^2\) — doubling \(n\) halves vs quarters
trap
- Tables have unequal widths. \(\Delta t\) comes from the \(t\)-values, never from dividing the range evenly.
- Midpoint is not exact — only usually better. Direction decides L/R; concavity decides M/T.
Take \(n\to\infty\) and the staircase becomes the exact signed area — the limit IS the definition.
Reading a limit as an integral: the constant inside the sample point is \(a\); the coefficient of \(k/n\) is the interval length, and over \(n\) it is \(\Delta x\).
trap
- Units multiply: integrand units \(\times\) \(dx\) units — (gal/min)·min = gal. The fastest answer-checker on the exam.
- Below the axis counts negative. "Area" questions and "integral" questions can differ.
Differentiating an accumulation undoes it: evaluate the integrand at the bound — no antidifferentiation happens.
trap
- The chain factor \(u'(x)\) is the most-forgotten point on this question type.
- A variable LOWER bound costs one minus sign: \(\int_x^a=-\int_a^x\).
Everything about \(g\) reads one derivative down on \(f\).
Graph-reading dictionary
sign of \(f\)direction of \(g\) (\(f>0\Rightarrow g\) rises)
\(f\) crosses \(+\to-\)local max of \(g\) (\(-\to+\): min)
\(f\) changes directioninflection of \(g\)
trap
- Slope vs value: where \(f\) peaks, \(g\) climbs fastest — not where \(g\) is biggest.
- \(g(a)=0\) is free — the first row of every candidates table.
Any antiderivative, evaluated at both ends. The \(+C\) cancels itself.
trap
- Both bounds, always — \(F(b)\) alone is the classic dropped point.
- \(\int_a^a=0\) needs no work, and \(\int_b^a=-\int_a^b\): bounds are oriented.
Twelve cards — every integration technique exists to reduce a stranger to one of these.
trap
- \(+C\) on every indefinite integral — a real rubric point.
- \(\ln|x|\), with the absolute value — it's what lets the antiderivative live on the negative side.
The chain rule, rewound: find a composition whose inside's derivative is standing right there.
The fingerprint check comes first, always: in \(\int x\,e^{x^2}dx\), the \(x\) is the fingerprint of \(x^2\). Constants adjust with \(du\) — the missing 3 in \(du=3x^2dx\) costs a \(\tfrac13\), never a 3.
trap
- Definite integrals convert the bounds through \(u(x)\) and finish entirely in u-world.
- No leftover x's — if an \(x\) survives the substitution, the u was wrong (or the technique is).
One quadratic-under-a-fraction, four fates. Run the checks in order.
The decision tree
top-heavy?long divide first — like \(\tfrac{17}{5}=3+\tfrac25\)
numerator \(\approx Q'\)?u-sub \(\to\ln|Q|\)
\(Q\) factors?partial fractions (6.12)
irreducible?complete the square \(\to\) arctan
trap
- The \(\tfrac1a\) out front is the most-dropped constant in the unit.
- The discriminant decides the branch: \(x^2-1\Rightarrow\) PF and logs; \(x^2+1\Rightarrow\) arctan.
The product rule, rewound — a trade, and the skill is trading down.
LIATE (Log, Inverse trig, Algebraic, Trig, Exponential): u = earliest (it dies under differentiation), dv = latest (it integrates forever). Tabular: polynomial dies → differentiate down, integrate down, signs \(+,-,+,-\), multiply the diagonals \(\searrow\). Cyclic (\(e^x\sin x\)): when the original returns, name it \(I\) and solve.
trap
- dv includes the dx — every time.
- Run the table to 0 — stopping at the constant silently drops the last term.
Un-adding a fraction: distinct linear factors split into deck cards.
Cover-up: to find \(A\), cover its factor and evaluate the survivors at that factor's root — one line per constant, no systems.
trap
- Top-light fractions only — divide first if improper.
- Check the fingerprint first: \(\int\!\tfrac{x}{x^2-4}dx\) looks like PF but is a one-line u-sub: \(\tfrac12\ln|x^2-4|\).
An ordinary integral plus a limit — swap the bad endpoint for a variable and let it run.
trap
- Write the limit — it's a scored step, and "diverges" is a complete answer.
- Scan for interior asymptotes: \(\int_{-1}^{1}x^{-2}dx=-2\) is the unit's most famous nonsense. Split at the blow-up; BOTH halves must converge.
- \(1/x\) is the anchor: it diverges in both regimes — \(p>1\) wins at \(\infty\), \(p<1\) wins near 0.
The unit's real final skill: five seconds of diagnosis before any computation.
The five-second scan
fingerprint pair?u-substitution — check this FIRST, even on products
unrelated product?parts (tabular if a polynomial dies)
rational?divide · ln · PF · arctan — run the 6.10 tree
\(\infty\) or a blow-up?limit first, then everything above applies inside it
trap
- Technique tunnel vision: just learned PF, so everything looks like PF. The scan order protects you.
- Techniques compose: \(\int_2^\infty\!\tfrac{dx}{x\ln^2 x}\) is a u-sub inside a limit.