Calculus 2 — Premium Practice Quiz

📚 Concept Review & Key Formulas

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u-substitution, integration by parts, trigonometric substitution, partial fractions, improper integrals

Area between curves, volume of revolution (disk/washer/shell), arc length, surface area, work

Convergence tests: ratio, root, integral, comparison, limit comparison, alternating series

Radius/interval of convergence, Taylor and Maclaurin series, term-by-term differentiation & integration

Parametric derivatives, arc length; polar coordinates, area in polar form

Separable ODEs, linear first-order (integrating factor), initial value problems

Integration by Parts

\(\displaystyle \int u\,dv = uv - \int v\,du\)

Geometric Series

\(\displaystyle \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r| < 1\)

Taylor Series (at \(x=a\))

\(\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)

Polar Area

\(\displaystyle A = \frac{1}{2}\int_{\alpha}^{\beta} [f(\theta)]^2\,d\theta\)

Arc Length (Cartesian)

\(\displaystyle L = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx\)

⭐ Must-Memorize Facts

Ratio Test: \(\lim|a_{n+1}/a_n|<1\) → converges; \(>1\) → diverges; \(=1\) → inconclusive
\(e^x=\sum x^n/n!\), \(\sin x=\sum(-1)^n x^{2n+1}/(2n+1)!\), \(\cos x=\sum(-1)^n x^{2n}/(2n)!\)
Disk method: \(V=\pi\int_a^b[f(x)]^2\,dx\); Shell method: \(V=2\pi\int_a^b x\,f(x)\,dx\)
Alternating Series Test: \(b_n\searrow 0\) → converges. Error \(\le b_{n+1}\)
\(\int \sec^2 x\,dx=\tan x+C\); \(\int \sec x\tan x\,dx=\sec x+C\); \(\int 1/(1+x^2)\,dx=\arctan x+C\)