CT2 — 16 Mar 2026 | Scope: 3.5–4.10 | Focus: Improper Integrals + Volume
Standard Antiderivatives
| $f(x)$ | $\int f(x)\,dx$ | $f(x)$ | $\int f(x)\,dx$ |
|---|---|---|---|
| $x^n$ $(n\neq -1)$ | $\frac{x^{n+1}}{n+1}$ | $\frac{1}{x}$ | $\ln|x|$ |
| $e^x$ | $e^x$ | $e^{-x}$ | $-e^{-x}$ |
| $\sin x$ | $-\cos x$ | $\cos x$ | $+\sin x$ |
| $\sec^2 x$ | $\tan x$ | $\csc^2 x$ | $-\cot x$ |
| $\frac{1}{1+x^2}$ | $\tan^{-1} x$ | $\frac{1}{\sqrt{1-x^2}}$ | $\sin^{-1} x$ |
Power rule trap: divide by NEW exponent $(n+1)$, not old $(n)$!
Must-Know Limits
| $\tan^{-1}(\infty) = \frac{\pi}{2}$ | $\tan^{-1}(0) = 0$ | $\tan^{-1}(-\infty) = -\frac{\pi}{2}$ |
| $e^{-\infty} = 0$ | $e^0 = 1$ | $\ln(0^+) = -\infty$ |
| $\frac{1}{\infty} = 0$ | $\frac{1}{0^+} = \infty$ | $\lim_{x\to 0^+} x\ln x = 0$ |
Improper Integrals (4.8)
Type I (infinite limits): $\int_a^\infty f,dx = \lim_{b\to\infty}\int_a^b f,dx$
Type II (discontinuity at endpoint $a$): $\int_a^b f,dx = \lim_{c\to a^+}\int_c^b f,dx$
Interior discontinuity at $d$: MUST split → $\int_a^d + \int_d^b$. Both must converge.
$p$-Test
| Converges | Diverges | |
|---|---|---|
| $\int_1^\infty \frac{1}{x^p},dx$ | $p > 1$ | $p \leq 1$ |
| $\int_0^1 \frac{1}{x^p},dx$ | $p < 1$ | $p \geq 1$ |
At $\infty$: need fast decay ($p>1$). At $0$: need mild blow-up ($p<1$). Directions flip!
Traps
- $\int_{-1}^1 \frac{1}{x^2},dx$ — looks finite but interior blow-up at $0$ → divergent
- Removable singularity ($\sin(x)/x \to 1$) → bounded → convergent
- $\ln$ singularity weaker than any $x^{-p}$ → typically convergent
Volume (4.9–4.10)
Formulas
| Method | Formula | When |
|---|---|---|
| Disk | $\pi\int_a^b [f(x)]^2,dx$ | Solid, no hole, $\perp$ to axis |
| Washer | $\pi\int_a^b [R^2 - r^2],dx$ | Two curves, hole in middle |
| Shell | $2\pi\int_a^b (\text{radius})(\text{height}),dx$ | Parallel to axis |
| Cross-section | $\int_a^b A(x),dx$ | Non-revolution |
Cross-Section Areas
| Shape | $A$ (side $s$) |
|---|---|
| Square | $s^2$ |
| Semicircle (diameter $s$) | $\frac{\pi}{8}s^2$ |
| Equil. triangle | $\frac{\sqrt{3}}{4}s^2$ |
Shifted Axis
Axis at $y = k$: radius $= f(x) - k$ (if axis below) or $k - f(x)$ (if axis above)
Axis at $x = k$: shell radius $= |x - k|$
Method Selection
| Axis | $y = f(x)$ given | Easier method |
|---|---|---|
| $x$-axis | Slice $dx$ $\perp$ axis | Disk/Washer |
| $y$-axis | Slice $dx$ $\parallel$ axis | Shell (avoids solving for $x$) |
| $y = k$ | Same as $x$-axis but shifted | Disk/Washer + shift |
| $x = k$ | Same as $y$-axis but shifted | Shell + shift |
Integration Techniques (Supporting)
IBP: $\int u,dv = uv - \int v,du$ — LIATE priority: Log > InvTrig > Algebra > Trig > Exp
Partial Fractions: $\frac{A}{x-a} \to A\ln|x-a|$ · $\frac{A}{(x-a)^n} \to \frac{A}{-(n-1)(x-a)^{n-1}}$
Power reduction: $\sin^2 u = \frac{1-\cos 2u}{2}$ · $\cos^2 u = \frac{1+\cos 2u}{2}$
Trig sub: $\sqrt{a^2+x^2} \to x=a\tan\theta$ · $\sqrt{a^2-x^2} \to x=a\sin\theta$ · $\sqrt{x^2-a^2} \to x=a\sec\theta$
Common Errors
| Error | Fix |
|---|---|
| Evaluate across interior discontinuity | ALWAYS scan for zeros in denominator → split |
| $\div$ old exponent in power rule | $\div$ NEW exponent $(n+1)$ |
| Swap default to $R_1 \leftrightarrow R_2$ | Check ALL rows before choosing swap target |
| $\int\cos x = -\sin x$ | That’s the derivative. $\int\cos x = +\sin x$ |
| Forgot shifted axis radius | Radius $= f(x) - k$, not $f(x)$ |
Full cheatsheet: CT2 Cheatsheet (3.5–4.10) · CT1 Cheatsheet