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§1 Integration Techniques
🧠 Key: LIATE — Log, Inverse trig, Algebraic, Trig, Exponential → pick u left-to-right
A physicist models the displacement of a damped oscillator as
\[ s(t) = \int_0^T t^3 e^{-2t}\,dt. \]
Using tabular (repeated) integration by parts, find the exact value of \(s(T)\) when \(T = 1\).
Which of the following equals \(\displaystyle\int_0^1 t^3 e^{-2t}\,dt\) ?
A
\(\dfrac{3}{8} - \dfrac{7}{8}e^{-2}\)
B
\(\dfrac{3}{8} - \dfrac{11}{8}e^{-2}\)
C
\(\dfrac{3}{4} - \dfrac{3}{4}e^{-2}\)
🧠 Key: COVER-UP — cover denominator factor, plug in its root → instant numerator
A biologist models a population with logistic growth. The integral that gives time \(t\) for the population \(P\) to grow from 200 to 800 (carrying capacity \(K = 1000\), rate \(r = 0.3\)) reduces to
\[\int_{200}^{800}\frac{1}{P(1000-P)}\,dP.\]
Evaluating this using partial fractions gives:
A
\(\dfrac{1}{1000}\ln\!\left(\dfrac{4}{1}\cdot\dfrac{200}{800}\right)\)
B
\(\dfrac{1}{500}\ln\!\left(\dfrac{P}{1000-P}\right)\Big|_{200}^{800}\)
C
\(\dfrac{1}{1000}\ln\!\left(\dfrac{P}{1000-P}\right)\Big|_{200}^{800} = \dfrac{\ln 16}{1000}\)
🧠 Key: SAT — √(a²–x²) → Sin, √(a²+x²) → Tan, √(x²–a²) → Sec
An engineer computes the arc length of a cable hanging between two towers. The arc length integral simplifies to
\[\int_0^3 \frac{x^2}{\sqrt{9-x^2}}\,dx.\]
Which value does this equal?
B
\(\dfrac{9\pi}{4} - 0 = \dfrac{9\pi}{4}\)
🧠 Key: P-TEST — ∫₁^∞ 1/xᵖ converges if p > 1; ∫₀¹ 1/xᵖ converges if p < 1 (opposite!)
A physicist integrates a probability density function:
\[I = \int_0^\infty x\,e^{-x^2/2}\,dx.\]
Students often confuse this with \(\int_{-\infty}^\infty e^{-x^2/2}\,dx = \sqrt{2\pi}\). Evaluate \(I\) exactly.
B
\(\dfrac{\sqrt{2\pi}}{2}\)
🧠 Key: ODD→SUB, EVEN→HALF-ANGLE — odd power? save one, sub u=cos/sin. even? use cos²x=(1+cos2x)/2
A signal processing engineer needs
\[\int_0^{\pi/2}\sin^4 x\,\cos^2 x\,dx.\]
What is the exact value?
§2 Applications of Integration
🧠 Key: PARALLEL→SHELL, PERP→DISK — axis parallel to strip → cylindrical shells; perpendicular → washers
The region \(R\) bounded by \(y = \sqrt{x}\), \(y = 0\), and \(x = 4\) is rotated about the
line \(x = 4\) (not the \(y\)-axis!).
Many students set up the shell method around \(x=0\) — a common error. Using shells about \(x=4\), the volume is:
🧠 Key: ARC = ∫√(1+[f′]²) dx — always square the derivative first, then add 1
A roller-coaster track follows \(y = \dfrac{x^2}{2} - \dfrac{\ln x}{4}\) for \(1 \le x \le e\).
The expression \(1+(y')^2\) simplifies beautifully into a perfect square — spot it and evaluate the arc length.
B
\(\dfrac{e^2}{2} + \dfrac{\ln e}{4} - \dfrac{3}{4} = \dfrac{e^2-1}{2}+\dfrac{1}{4}\)
🧠 Key: W = ∫F dx = ∫kx dx — natural length doesn't matter for work, only displacement matters
A spring has natural length 30 cm. A force of 20 N is required to stretch it to 45 cm.
How much work (in joules) is done in stretching the spring from 35 cm to 50 cm?
🧠 Key: MOMENT/MASS — x̄ = Mᵧ/m, ȳ = Mₓ/m; moment arm = distance from axis
A thin plate with uniform density \(\rho\) occupies the region bounded by \(y = x^2\) and \(y = x\).
Find \(\bar{y}\), the \(y\)-coordinate of the centroid (density cancels out).
🧠 Key: F = ρg∫depth · width dy — pressure increases with depth; integrate horizontal strips
A vertical trapezoidal dam gate has a top width of 6 m, a bottom width of 2 m, and a height of 4 m. Water fills to the top.
Using \(\rho g = 9800\) N/m³, set up and evaluate the total hydrostatic force on the gate (take depth \(y\) measured downward from the water surface).
§3 Sequences, Series & Convergence
🧠 Key: RATIO — L = lim|aₙ₊₁/aₙ|; L<1 converge, L>1 diverge, L=1 inconclusive (try another test!)
A student claims the series \(\displaystyle\sum_{n=1}^{\infty}\frac{n!\,3^n}{n^n}\) converges.
Apply the Ratio Test. Note: \(\lim_{n\to\infty}\!\left(1+\frac{1}{n}\right)^n = e\). What does the Ratio Test yield?
A
Converges; \(L = \dfrac{3}{e} < 1\)
C
Converges; \(L = \dfrac{3}{e} \approx 1.10 > 1\) — Wait, diverges!
D
Diverges; \(L = e/3 > 1\)
🧠 Key: AST-ERROR ≤ |aₙ₊₁| — error is bounded by the FIRST omitted term's absolute value
A computer scientist approximates \(\ln 2\) using the alternating harmonic series:
\[\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots\]
What is the minimum number of terms needed to guarantee the approximation is within \(0.001\) of \(\ln 2\)?
🧠 Key: R = 1/L where L = lim|aₙ₊₁/aₙ| — always CHECK endpoints separately!
Find the interval of convergence (including endpoint checks) of
\[\sum_{n=1}^{\infty} \frac{(-1)^n (x-2)^n}{n\cdot 3^n}.\]
At \(x = 5\): harmonic series (diverge); at \(x = -1\): alternating harmonic (converge). Classify correctly.
A
\((-1, 5)\) (open interval)
B
\([-1, 5)\) (closed left, open right)
C
\((-1, 5]\) (open left, closed right)
D
\([-1, 5]\) (closed interval)
🧠 Key: SUBSTITUTE-THEN-INTEGRATE — derive new series by subbing into known ones, never differentiate from scratch
Using the Maclaurin series for \(e^x\), find the sum of the series
\[\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!} = 1 - \frac{1}{6} + \frac{1}{120} - \cdots\]
Hint: recognize this as a familiar trig value evaluated at a specific input.
🧠 Key: LCT-DOMINATE — Limit Comparison: pick bₙ that "looks like" aₙ for large n, ignore constants
Determine whether \(\displaystyle\sum_{n=1}^\infty \frac{\sqrt{n}+\sin n}{n^2+\cos n}\) converges or diverges.
The oscillating terms \(\sin n\) and \(\cos n\) confuse many students into thinking they need the AST. Use the Limit Comparison Test instead.
A
Diverges by Comparison with \(\sum \frac{1}{\sqrt{n}}\)
B
Inconclusive — AST required
C
Converges; LCT with \(\sum n^{-3/2}\) gives limit 1
D
Diverges by the Divergence Test
§4 Differential Equations · Parametric · Polar
🧠 Key: SEPARATE-dy/dx → ∫f(y)dy = ∫g(x)dx — always solve for C using initial condition AFTER integrating
A tank holds 500 L of salt water with 10 kg of salt. Pure water enters at 5 L/min, and well-mixed solution exits at 5 L/min.
The amount of salt \(A(t)\) (kg) satisfies \(\dfrac{dA}{dt} = -\dfrac{A}{100}\). Find \(A(t)\) and determine when the salt drops below 1 kg.
A
\(t = 100\ln 5 \approx 161\) min
B
\(t = 100\ln 10 \approx 230\) min
C
\(t = 50\ln 10 \approx 115\) min
D
\(t = 200\ln 2 \approx 139\) min
🧠 Key: A = ∫y(dx/dt)dt — parametric area: replace dx with (dx/dt)dt, watch the direction!
A particle traces the curve \(x = t - \sin t\), \(y = 1 - \cos t\) for \(0 \le t \le 2\pi\) (one arch of a cycloid).
Find the area under one arch of the cycloid (above the \(x\)-axis).
🧠 Key: A = ½∫(r_outer² – r_inner²)dθ — find intersection angles first; then outer minus inner
Find the area of the region inside the circle \(r = 3\sin\theta\) and outside the cardioid \(r = 1 + \sin\theta\).
Many students forget to find where the curves intersect (\(r = 3\sin\theta = 1+\sin\theta \Rightarrow \sin\theta = \tfrac{1}{2}\)) and use wrong limits.
🧠 Key: 1^∞ → take LN → L'Hop → exponentiate — the sneakiest indeterminate form; always ln both sides first
Evaluate
\[\lim_{x\to 0^+} \left(1 + \sin 4x\right)^{\cot x}.\]
This is type \(1^\infty\). Take \(\ln\), convert to \(0/0\), apply L'Hôpital, then exponentiate. Students often forget the final exponentiation step.
🧠 Key: |Rₙ| ≤ M·|x-a|ⁿ⁺¹/(n+1)! where M = max of |f⁽ⁿ⁺¹⁾| on the interval
Approximate \(\sqrt{e}\) using the 3rd-degree Maclaurin polynomial for \(e^x\) evaluated at \(x = \tfrac{1}{2}\):
\[P_3\!\left(\tfrac{1}{2}\right) = 1 + \frac{1}{2} + \frac{1}{8} + \frac{1}{48}.\]
Use the Lagrange Remainder to find an upper bound on the error \(|e^{1/2} - P_3(1/2)|\). Note: \(e^{1/2} < 2\), so bound \(f^{(4)}(c) = e^c \le 2\) for \(c\in(0,\tfrac{1}{2})\).
B
\(\dfrac{1}{192}\) — same! Error \(\le \dfrac{2\cdot(1/2)^4}{4!} = \dfrac{1}{192}\)
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