Volumes by Cylindrical Shells

6. 3: Volumes by Cylindrical Shells

• Cylindrical shells are used to calculate volume when rotating a region around an axis.
• The disk method can be problematic, such as when solving for x in terms of y

Cylindrical Shells Explained

• V = (area of outer cylinder) - (area of inner cylinder)
• Area of outer cylinder = $\pi r_2^2h$
• Area of inner cylinder = $\pi r_1^2h$
• $V = \pi h(r_2^2 - r_1^2)$
• $V = 2\pi h \left( \frac{r_2 + r_1}{2} \right) (r_2 - r_1)$

Formula Variables

• $\Delta r = r_2 - r_1$
• $r = \frac{r_2 + r_1}{2}$
• $V = 2\pi rh \Delta r$

General formula for volume of cylindrical sells

• $V = \int_a^b 2\pi x f(x) dx$
• It is the region under $y = f(x)$ from a to b rotated about the y-axis

Example 1

• The region under the curve $y = 2x^2 - x^3$ from $0$ to $2$ is rotated about the y-axis. Solution:
• $V = \int_0^2 2\pi x (2x^2 - x^3) dx$
• $2\pi \int_0^2 (2x^3 - x^4) dx$
• $2\pi \left[ \frac{1}{2}x^4 - \frac{1}{5}x^5 \right]_0^2$
• $2\pi \left( \frac{1}{2}(2)^4 - \frac{1}{5}(2)^5 \right)$
• $2\pi \left( 8 - \frac{32}{5} \right)$
• $2\pi \left( \frac{40 - 32}{5} \right)$
• $2\pi \left( \frac{8}{5} \right) = \frac{16\pi}{5}$

Example 2

• The region bounded by $y = x - x^2$ and $y = 0$ is rotated about the line $x = 2$. Solution:
• $V = \int_0^1 2\pi (2 - x)(x - x^2) dx$
• $2\pi \int_0^1 (2x - 2x^2 - x^2 + x^3) dx$
• $2\pi \int_0^1 (2x - 3x^2 + x^3) dx$
• $2\pi \left[ x^2 - x^3 + \frac{1}{4}x^4 \right]_0^1$
• $2\pi \left( 1 - 1 + \frac{1}{4} \right)$
• $2\pi \left( \frac{1}{4} \right) = \frac{\pi}{2}$

Example 3

• The region enclosed by $y = x$ and $y = x^2$ is rotated about the x-axis. Solution:
• $V = \int_0^1 2\pi y (\sqrt{y} - y) dy$
• $2\pi \int_0^1 (y^{3/2} - y^2) dy$
• $2\pi \left[ \frac{2}{5}y^{5/2} - \frac{1}{3}y^3 \right]_0^1$
• $2\pi \left( \frac{2}{5} - \frac{1}{3} \right)$
• $2\pi \left( \frac{6 - 5}{15} \right)$
• $2\pi \left( \frac{1}{15} \right) = \frac{2\pi}{15}$

Algorithmes gloutons

• A simple technique for tackling optimization problems, by picking the current best option for the desired endpoint.
• Series of choices, each seemingly the best at that moment
• These choices hopefully to lead to an overall optimal solution.

Operation

• Define problem, identify the objective
• Choose best local option - most promising without worrying about future consequences
• Step by step, repeat to complete a solution

Advantages

• Simple, easy to understand and implement
• Efficient, able to make direct decisions

Disadvantages

• Non-optimality, does not guarantee the best solution
• Need for proof, greedy approach needs 3rd party verification

Rendu de monnaie

• Return a sum of money with the least ammount of bills and coins as possible
• Algorithm: Sort denominations by value, take as much highest value without passing the sum
• Take the next smallest bill repeat

Problème du sac à dos fractionnaire

• Maximizes the total value of objects carried in a backpack with limited capacity, where fractions of objects can be taken. Algorithm:
• Calculate the value to wheight ratio, sort objects by descending ratio.
• Take as much of the highest ratio as possible, repeating process
• Greedy algorithm is optimal for the fractional knapsack calculation

Usage Considerations

• Optimal local structure - a choice at one state leads to global solution
• Solution is needed quickly
• As a precursor to other complex algorithms

Reglas de la inferencia

• Logical step in reasoning process, moves from premise to conclusion. Guarantees true premise will be met with true conlcusion.

Modus Ponens (MP)

• Forma:
• P → Q
• P
• ∴ Q
• Ejemplo:
• Si está lloviendo, entonces hay nubes en el cielo.
• Está lloviendo.
• ∴ Hay nubes en el cielo.

Modus Tollens (MT)

• Forma:
• P → Q
• ¬ Q
• ∴ ¬ P
• Ejemplo:
• Si está lloviendo, entonces hay nubes en el cielo.
• No hay nubes en el cielo.
• ∴ No está lloviendo.

Silogismo Hipotético (SH)

• Forma:
• P → Q
• Q → R
• ∴ P → R
• Ejemplo:
• Si estudio, entonces obtendré buenas notas.
• Si obtengo buenas notas, entonces entraré a la universidad.
• ∴ Si estudio, entonces entraré a la universidad.

Silogismo Disyuntivo (SD)

• Forma:
• P ∨ Q
• ¬ P
• ∴ Q
• Ejemplo:
• O voy al cine, o me quedo en casa.
• No voy al cine.
• ∴ Me quedo en casa.

Adición (Ad)

• Forma:
• P
• ∴ P ∨ Q
• Ejemplo:
• Estoy leyendo un libro.
• ∴ Estoy leyendo un libro o estoy viendo una película.

Simplificación (Simp)

• Forma:
• P ∧ Q
• ∴ P
• Ejemplo:
• Estoy leyendo un libro y escuchando música.
• ∴ Estoy leyendo un libro.

Conjunción (Conj)

• Forma:
• P
• Q
• ∴ P ∧ Q
• Ejemplo:
• Estoy leyendo un libro.
• Estoy escuchando música.
• ∴ Estoy leyendo un libro y escuchando música.

Ley de Morgan (LM)

• Forma:
• ¬ (P ∧ Q) ≡ (¬ P) ∨ (¬ Q)
• ¬ (P ∨ Q) ≡ (¬ P) ∧ (¬ Q)
• Ejemplo:
• No es cierto que estoy leyendo un libro y escuchando música ≡ No estoy leyendo un libro o no estoy escuchando música.No es cierto que estoy leyendo un libro o escuchando música ≡ No estoy leyendo un libro y no estoy escuchando música.

Doble Negación (DN)

• Forma:
• P ≡ ¬ ¬ P
• Ejemplo:
• No es cierto que no estoy leyendo un libro ≡ Estoy leyendo un libro.

Conmutación (Conm)

• Forma:
• (P ∨ Q) ≡ (Q ∨ P)
• (P ∧ Q) ≡ (Q ∧ P)
• Ejemplo:
• Estoy leyendo un libro o escuchando música ≡ Estoy escuchando música o estoy leyendo un libro.
• Estoy leyendo un libro y escuchando música ≡ Estoy escuchando música y estoy leyendo un libro.

Implicación Material (IM)

• Forma:
• P → Q ≡ ¬ P ∨ Q
• Ejemplo:
• Si estoy leyendo un libro, entonces estoy aprendiendo ≡ No estoy leyendo un libro o estoy aprendiendo.This provides fundamental tools for the construction of logical arguments and mathematical theorems.

Physics - Vectors

• Vectors have magnitude, direction, and sense, representing physical quantities.

Vector Components

• Expressed as $\vec{a} = (a_x, a_y)$ where:
• $a_x = a \cos \theta$
• $a_y = a \sin \theta$
• And $\vec{a} = a_x \hat{i} + a_y \hat{j}$

Vector Magnitude

• Calculated as $a = |\vec{a}| = \sqrt{a_x^2 + a_y^2}$

Vector Addition/ Subtraction

• Sum $\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y)$

Scalar Product

• Calculated as $\vec{a} \cdot \vec{b} = a b \cos \alpha = a_x b_x + a_y b_y$

Vector Product

• Calculated as $\vec{a} \times \vec{b} = a b \sin \alpha \hat{k}$

Kinematics - Uniform Rectilinear Motion (MRU)

• Formula $v = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$
• Position $x(t) = x_i + v (t - t_i)$

Uniformly Varied Rectilinear Motion (MRUV)

• Formula $a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$
• Velocity $v(t) = v_i + a (t - t_i)$
• Position $x(t) = x_i + v_i (t - t_i) + \frac{1}{2} a (t - t_i)^2$
• Alternate Formula $v_f^2 - v_i^2 = 2 a \Delta x$

Vertical Shot and Free Fall

• Special cases of MRUV where acceleration is due to gravity ($g = 9,8 m/s^2$) and the path is vertical.

Oblique Shot

• Composition of MRU on the x-axis and MRUV on the y-axis.
• $x(t) = x_i + v_{ix} (t - t_i)$
• $v_x(t) = v_{ix}$
• $y(t) = y_i + v_{iy} (t - t_i) + \frac{1}{2} a_y (t - t_i)^2$
• $v_y(t) = v_{iy} + a_y (t - t_i)$

Circular Motion

• $v_{ix} = v_i \cos \theta$
• $v_{iy} = v_i \sin \theta$
• $a_y = -g$

Uniform Circular Motion (MCU)

• Angular Velocity $\omega = \frac{\Delta \theta}{\Delta t}$
• Angular Position $\theta(t) = \theta_i + \omega (t - t_i)$
• Linear Velocity $v = \omega r$
• Centripetal Acceleration $a_c = \frac{v^2}{r} = \omega^2 r$
• Period $T = \frac{2 \pi r}{v} = \frac{2 \pi}{\omega}$

Uniformly Varied Circular Motion (MCUV)

• Angular Acceleration $\alpha = \frac{\Delta \omega}{\Delta t}$
• Angular Velocity $\omega (t) = \omega_i + \alpha (t - t_i)$
• Angular Position $\theta (t) = \theta_i + \omega_i (t - t_i) + \frac{1}{2} \alpha (t - t_i)^2$

Dynamics - Newton's Laws

• First Law: Inertia - objects stay at rest unless acted on by an external force.
• Second Law: $\sum \vec{F} = m \vec{a}$, where force = mass * acceleration
• Third Law: Action-Reaction - every action has an equal and opposite reaction.

Work

• Formula $W = \vec{F} \cdot \Delta \vec{x} = F \Delta x \cos \theta$

Kinetic Energy

• Formula $K = \frac{1}{2} m v^2$

Work-Energy Theorem

• Formula $W_{neto} = \Delta K$

Potential Energy

• Gravitational Potential Energy: $U_g = m g h$
• Elastic Potential Energy: $U_e = \frac{1}{2} k x^2$

Mechanical Energy

• $E = K + U$

Conservative Forces

• Forces where work done is independent of the path (e.g., gravity).

Conservation of Mechanical Energy

• If only conservative forces act, $\Delta E = 0$ and $E_i = E_f$

Power

• Power = $\frac{\Delta W}{\Delta t} = \vec{F} \cdot \vec{v}$

Momentum

• Momentum: $\vec{p} = m \vec{v}$

Impulse

• Impulse: $\vec{I} = \vec{F} \Delta t = \Delta \vec{p}$

Conservation of Momentum

• No external force keeps momentum constant ($\Delta \vec{p} = 0$ and $\vec{p}_i = \vec{p}_f$)

Collisions

• Elastic: Conserves kinetic energy.
• Inelastic: Does not conserve kinetic energy.
• Perfectly Inelastic: Objects stick together.

Statics - Equilibrium Conditions

• Force = $\sum \vec{F} = 0$
• Torque = $\sum \tau = 0$

Chemical Kinetics

• Reaction Rate
• Reaction rate expresses how fast the reactants turn into product
• It is the change in concentration of a reactant or product per unit time with the formula Rate = $-\frac{\Delta[Reactant]}{\Delta t} = \frac{\Delta[Product]}{\Delta t}$
• Example $2H_2(g) + O_2(g) \rightarrow 2H_2O(g)$ turns to $rate = -\frac{1}{2}\frac{\Delta[H_2]}{\Delta t} = -\frac{\Delta[O_2]}{\Delta t} = \frac{1}{2}\frac{\Delta[H_2O]}{\Delta t}$

Factors Affecting Reaction Rate

• Concentration
• Temperature
• Surface Area
• Catalyst
• $aA + bB \rightarrow Products$ turns to Rate = $k[A]^m[B]^n$
• k is the rate constant
• $[A]$ and $[B]$ are the concentrations of reactants
• m and n are the reaction orders
• Examples
  • Zero order
  • First order
  • Second order
• Rate law follows equation $Rate = k[NO]^2[Cl_2]$

Reaction Mechanisms

• Reaction mechanism is the step by step process of overall change with elementary reaction and its molecularity number.

Activation Energy

• It is the minimum amount of energy required for a reaction to occur with Arrhenius Equation listed below:
• $k = Ae^{-\frac{E_a}{RT}}$
• k is the rate constant.
• A is the frequency factor.
• Ea is the activation energy.
• R is the ideal gas constant ($8.314 J / (mol \cdot K)$
• T is the temperature in Kelvin

Catalysis

• A catalyst speeds up with out being consumed
  • Homogeneous Catalyst
  • Heterogeneous Catalyst
• $2H_2O_2(aq) \xrightarrow{I^-(aq)} 2H_2O(l) + O_2(g)$

Algèbre linéaire, Cours et exercices corrigés Book Information

• Claude Descharnps - Professeur de mathématiques spéciales au lycée Louis-le-Grand
• Alain Warusfel - Professeur de mathématiques spéciales au lycée Louis-le-Grand
• Anne-Marie Décaillot colaboration

Table des matières

• Chapitre 1: Espaces vectoriels
• Chapitre 2: Espaces vectoriels de dimension finie
• Chapitre 3: Matrices
• Chapitre 4: Déterminants
• Chapitre 5: Réduction des endomorphismes
• Chapitre 6: Espaces préhilbertiens réels
• Chapitre 7: Annexes

Exercices ( extraits)

• $Soient E un espace vectoriel sur \mathbb{R}, et $p$ et $q$ deux projecteurs of E tels que: $$p + q = id_E$$. Montrer que: $$E = Im , p $ \oplus $ Im , q
• Soient$E$ un espace vectoriel, $f \in \mathcal{L}(E)$ et $x_0 \in$ E, on considère l'application:
• Soit$A = \begin{pmatrix} 1 & 1 & -1 \ 1 & 0 & 0 \ -1 & 0 & 0 \end{pmatrix}$. Calculer $A^n$ pour tout$n \in \mathbb{N}$.
• Calculer le déterminant de la matrice \begin{pmatrix} A = (a_(ij)\end{pmatrix} \in M_n(\mathbb{R})$ définie par$ a_(ij) = \underset{}^{} cos(i+j)$
• Soit $u\in (E) tel que u^3 = u^2 + u - . Монtrer cue uest inversible et exprimer ${u}^{-1}$ en fonction de u.

Formulario de errores comunes

Cinemática

• Posición: $x(t) = x_0 + v_0.t + 1/2.a.t^2$
• Velocidad: $V(t) = V_0 + at$
• Aceleración: $a(t) = a$

Dinámica

• Fuerza gravitatoria: $Fg = m.g$
• Fuerza elástica: $Fe = -k.x$
• Fuerza de rozamiento: $Fr = \mu.N$

Trabajo y energía

• Energía cinética: $Ec = 1/2.m.v^2$
• Energía potencial gravitatoria: $Epg = m.g.h$
• Energía potencial elástica: $Epe = 1/2.k.x^2$
• Teorema trabajo-energía: $W = \Delta Ec + \Delta Epg + \Delta Epe$

Fluidos

• Diversas formulas including
• Densidad: $\rho = m/V$
• Presión: $P = F/A$
• Principio de Arquímedes: $E = \rho.Vg$
• Caudal: $Q = A.V$

Termodinámica

• Calor sensible: $Q = m.c.\Delta T$
• Calor latente: $Q = m.L$
• Primer principio de la termodinámica: $\Delta U = Q - W$
• Rendimiento de una máquina térmica: $\eta = W/Qc$

Electromagnetismo

Several formulas including

• Ley de Coulomb: $F = k.(q1.q2)/d^2$
• Campo eléctrico: $E = F/q$
• Potencial eléctrico: $V = k.q/d$
• Capacidad: $C = Q/V$
• Resistencia: $R = V/I$
• Ley de Ohm: $V = I.R$
• Potencia eléctrica: $P = V.I$

Física cuántica

• Energía de un fotón: $E = h.f$
• Longitud de onda de De Broglie: $\lambda = h/p$
• Principio de incertidumbre de Heisenberg: $\Delta x.\Delta p >= h/4\pi$

Explanation of using cylindrical shells to calculate volume. Includes the general formula for volume using cylindrical shells. Includes a worked example of a region rotated around the y-axis