Integral Study Guide

📖 Core Concepts Definite integral – the signed area under \(f(x)\) from \(a\) to \(b\); written \(\displaystyle\int{a}^{b} f(x)\,dx\). Indefinite integral – the family of antiderivatives of \(f\); written \(\displaystyle\int f(x)\,dx = F(x)+C\). Integrand – the function being integrated, \(f(x)\). Limits of integration – the numbers \(a\) and \(b\) that bound the interval \([a,b]\). Fundamental Theorem of Calculus (FTC) – links integration and differentiation: \(F(x)=\int{a}^{x} f(t)\,dt \;\Rightarrow\; F'(x)=f(x)\). If \(F'=f\), then \(\displaystyle\int{a}^{b} f(x)\,dx = F(b)-F(a)\). Riemann integral – limit of Riemann sums \(\sum f(ti)\Delta xi\) as the partition gets arbitrarily fine. Improper integral – defined by a limit when the interval or integrand is unbounded. 📌 Must Remember Linearity: \(\displaystyle\int{a}^{b} (c\,f+g)\,dx = c\int{a}^{b} f\,dx + \int{a}^{b} g\,dx\). Order preservation: If \(f\le g\) on \([a,b]\), then \(\int{a}^{b} f \le \int{a}^{b} g\). Bounding: \(m(b-a) \le \int{a}^{b} f \le M(b-a)\) when \(m\le f\le M\). Absolute‑value inequality: \(\big|\int{a}^{b} f\big| \le \int{a}^{b} |f|\). Limit‑swap sign: \(\int{a}^{b} = -\int{b}^{a}\); if \(a=b\), integral = 0. Disc method (volume): \(V = \pi\displaystyle\int{a}^{b} [f(x)]^{2}\,dx\). Shell method (volume): \(V = 2\pi\displaystyle\int{a}^{b} x\,f(x)\,dx\). Work: \(W = \displaystyle\int{a}^{b} F(x)\,dx\). Displacement: \(\displaystyle\int{t0}^{t1} v(t)\,dt\). 🔄 Key Processes Evaluating a definite integral (FTC, part 2) Find an antiderivative \(F\) of \(f\). Compute \(F(b)-F(a)\). Riemann‑sum approximation Partition \([a,b]\) into \(n\) subintervals, \(\Delta xi = xi-x{i-1}\). Choose sample points \(ti\). Form \(\displaystyle Sn=\sum{i=1}^{n} f(ti)\Delta xi\). Let \(n\to\infty\); \(Sn\to\int{a}^{b} f\). Improper integral evaluation Replace infinite bound or singular endpoint with a parameter (e.g., \(M\) or \(\epsilon\)). Compute the proper integral, then take the limit \(M\to\infty\) or \(\epsilon\to0^{+}\). Double integral via Fubini Write \(\displaystyle\iintR f(x,y)\,dA = \int{a}^{b}\!\big(\int{c}^{d} f(x,y)\,dy\big)dx\). Integrate inner variable first, then outer. 🔍 Key Comparisons Definite vs. Indefinite Integral Definite: limits present → numeric value (signed area). Indefinite: no limits → antiderivative + constant \(C\). Improper vs. Proper Integral Proper: integrand finite on a closed, bounded interval. Improper: at least one bound infinite or integrand singular; defined via limits. Rectangle, Trapezoidal, Simpson’s Rules Rectangle: uses left/right endpoints → \(O(\Delta x)\) error. Trapezoidal: averages endpoints → \(O(\Delta x^{2})\) error. Simpson’s: fits quadratics → \(O(\Delta x^{4})\) error (exact for degree ≤ 3). ⚠️ Common Misunderstandings “Integral equals area” – only true when \(f(x)\ge0\); negative parts subtract. Switching limits without a sign change – forgetting \(\int{a}^{b} = -\int{b}^{a}\). Assuming any antiderivative works – the FTC requires \(F\) to be defined on the whole interval \([a,b]\). Improper integrals always converge – many diverge; always evaluate the limit. 🧠 Mental Models / Intuition Signed area picture: Imagine piling thin strips of height \(f(x)\) and width \(dx\); strips above the axis add, those below subtract. Riemann sum → limit: Think of a video game where you increase resolution; the picture (integral) becomes smoother as pixel size (\(\Delta x\)) shrinks. FTC as a “reverse derivative”: Differentiating undoes the accumulation of area; integrating “undoes” differentiation. 🚩 Exceptions & Edge Cases Discontinuous integrand: Riemann integrable if the set of discontinuities has measure zero (e.g., finite jumps). Zero‑width interval: \(\int{a}^{a} f = 0\) even if \(f\) is undefined at \(a\). Improper integrals with oscillatory behavior: Convergence may depend on principal value (not covered in basic outline). 📍 When to Use Which Exact value needed: Use FTC (find antiderivative) or analytic techniques (substitution, parts, partial fractions). Integrand messy but integrable numerically: Choose Simpson’s rule for smooth functions; trapezoidal for piecewise‑linear approximations; Gaussian quadrature for high accuracy with few points. Unbounded domain or singularity: Set up an improper integral and evaluate the limit. Multidimensional region is a rectangle: Apply Fubini → iterated single integrals. 👀 Patterns to Recognize \(f'(x)\) inside the integral → try substitution \(u = f(x)\). Product of a function and its derivative → integration by parts often simplifies. Rational function with degree numerator ≥ denominator → perform polynomial division first, then partial fractions. Even/odd symmetry on \([-a,a]\) → integral of odd function = 0; even function = 2× integral from 0 to \(a\). Repeated quadratic forms → consider trigonometric substitution (e.g., \(\sqrt{a^{2}-x^{2}}\)). 🗂️ Exam Traps Choosing wrong sign when swapping limits – answer will be off by a minus sign. Treating an improper integral as proper – missing the limit step leads to “convergent” false positives. Applying Simpson’s rule on an odd number of subintervals – Simpson requires an even number of subintervals; using odd gives a wrong formula. Forgetting the constant \(C\) in indefinite integrals – on multiple‑choice, “+ C” is often required. Mixing up disc vs. shell method formulas – disc uses \(\pi f^{2}\) (radius squared), shell uses \(2\pi x f\) (radius × height). --- Keep this sheet handy; the bullet format makes quick scanning before the exam painless.