Calculus Integration Techniques and Applications Study Guide

Prerequisite Integration Knowledge and Basic Antiderivatives

• Essential Prerequisite Trigonometric Values:

  • \n\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\n

  • \n\arctan(1) = \frac{\pi}{4}\n

• Fundamental Table of Antiderivatives:

  • Power Rule for Integrals:

  • \n\int x^n \, dx = \frac{x^{n+1}}{n + 1} + C \quad \text{for } n \neq -1\n

  • Reciprocal Rule:

  • \n\int \frac{1}{x} \, dx = \ln|x| + C\n

  • Exponential Functions:

  • \n\int e^x \, dx = e^x + C\n

  • \n\int a^x \, dx = \frac{a^x}{\ln(a)} + C \quad \text{for } a > 0, a \neq 1\n

  • Trigonometric Functions:

  • \n\int \cos(x) \, dx = \sin(x) + C\n

  • \n\int \sin(x) \, dx = -\cos(x) + C\n

  • \n\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C\n

  • \n\int \csc(x) \, dx = -\ln|\csc(x) + \cot(x)| + C\n

  • \n\int \tan(x) \, dx = \ln|\sec(x)| + C\n

  • \n\int \cot(x) \, dx = \ln|\sin(x)| + C\n

  • \n\int \sec^2(x) \, dx = \tan(x) + C\n

  • \n\int \csc^2(x) \, dx = -\cot(x) + C\n

  • \n\int \sec(x)\tan(x) \, dx = \sec(x) + C\n

  • \n\int \csc(x)\cot(x) \, dx = -\csc(x) + C\n

  • Inverse Trigonometric Integrals:

  • \n\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C\n

  • \n\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C\n

  • \n\int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| + C\n

Integration by Substitution Method

• Substitution Method for Indefinite Integrals:

  • Definition and Formula:

  • \n\int f(g(x))g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x)\n

  • Mandatory Procedural Requirements:

  • Explicitly state the choice for $u$.

  • Show the correct computation of the differential $du = g'(x) \, dx$.

  • Convert the entire integrand to be in terms of $u$.

  • After computing the integral in terms of $u$, substitute $g(x)$ back in place of $u$ so the final expression is written purely in terms of the original variable $x$.

• Substitution Method for Definite Integrals:

  • Definition and Formula:

  • \n\int_{a}^{b} f(g(x))g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du \quad \text{where } u = g(x)\n

  • Limit Transformation Requirements:

  • Lower limit $x = a$ transforms to $u = g(a)$.

  • Upper limit $x = b$ transforms to $u = g(b)$.

  • Limits of integration must be explicitly converted and written in terms of $u$.

Calculating Areas of Plane Regions

• Area Integration with Respect to $x$:

  • Formula for a region bounded above by $y = f(x)$ and below by $y = g(x)$ on interval $[a, b]$:

  • \n\text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx\n

  • Visual Representation:

  • 
    

  

• Area Integration with Respect to $y$:

  • Formula for a region bounded on the right by $x = f(y)$ and on the left by $x = g(y)$ on interval $[c, d]$:

  • \n\text{Area} = \int_{c}^{d} [f(y) - g(y)] \, dy\n

  • Visual Representation:

  • 
    

  

• Multi-Integral Regions and Problem Formats:

  • Regions Requiring a Sum of Integrals:

  • When boundary functions cross or change positions within the interval, the region must be split into subregions, requiring a sum of integrals with respect to $x$ or $y$.

  • Setup vs. Evaluation Requirements:

  • Setting up an integral requires writing a definite integral with explicit limits and integrand.

  • Evaluating an integral requires calculating the integration to arrive at a single numeric value.

Volume Evaluation using Cross-Sections

• General Volume Formula by Parallel Cross-Sections:

  • Volume equation for cross-sections perpendicular to the $x$-axis:

  • \n\text{Volume} = \int_{a}^{b} A(x) \, dx\n

  • $A(x)$ represents the cross-sectional area of the solid cut by a plane perpendicular to the $x$-axis at position $x$.

  • For cross-sections perpendicular to the $y$-axis or $z$-axis, volume is similarly given by \n\int_{c}^{d} A(y) \, dy\n or \n\int A(z) \, dz\n.

  • Visual Representation:

  • 
    

  

Volumes of Solids of Revolution: Washer Method

• Washer Method Mechanics:

  • Formed by revolving rectangular strips perpendicular to the axis of revolution.

  • Generates cross-sections shaped like concentric circular rings (washers).

  • The Disk Method is a special case of the Washer Method where the inner radius $r = 0$, occurring when the axis of revolution forms a boundary of the region.

• Washer Formula for Horizontal Axes of Revolution:

  • Revolves vertical strips, requiring integration with respect to $x$.

  • Boundary functions expressed as $y = f(x)$.

  • Formula:

  • \n\text{Volume} = \int_{a}^{b} \pi \left( (R(x))^2 - (r(x))^2 \right) \, dx\n

  • $R(x)$: Outer radius (distance between axis of revolution and outer boundary curve).

  • $r(x)$: Inner radius (distance between axis of revolution and inner boundary curve).

  • Visual Representation:

  • 
    

  

• Washer Formula for Vertical Axes of Revolution:

  • Revolves horizontal strips, requiring integration with respect to $y$.

  • Boundary functions expressed as $x = g(y)$.

  • Formula:

  • \n\text{Volume} = \int_{c}^{d} \pi \left( (R(y))^2 - (r(y))^2 \right) \, dy\n

  • $R(y)$: Outer radius from vertical axis to outer curve.

  • $r(y)$: Inner radius from vertical axis to inner curve.

  • Visual Representation:

  • 
    

  

Volumes of Solids of Revolution: Shell Method

• Cylindrical Shell Method Mechanics:

  • Formed by revolving rectangular strips parallel to the axis of revolution.

  • Generates thin cylindrical shells centered around the axis of revolution.

• Shell Formula for Vertical Axes of Revolution:

  • Revolves vertical strips, requiring integration with respect to $x$.

  • Boundary functions expressed as $y = f(x)$.

  • Formula:

  • \n\text{Volume} = \int_{a}^{b} 2\pi r(x) h(x) \, dx\n

  • $r(x)$: Shell radius (distance between vertical axis of revolution and strip at $x$).

  • $h(x)$: Shell height (length/height of vertical strip at $x$).

  • Visual Representation:

  • 
    

  

• Shell Formula for Horizontal Axes of Revolution:

  • Revolves horizontal strips, requiring integration with respect to $y$.

  • Boundary functions expressed as $x = g(y)$.

  • Formula:

  • \n\text{Volume} = \int_{c}^{d} 2\pi r(y) h(y) \, dy\n

  • $r(y)$: Shell radius (distance between horizontal axis of revolution and strip at $y$).

  • $h(y)$: Shell height (length/width of horizontal strip at $y$).

  • Visual Representation:

  • 
    

  

• Method Comparison and Selection:

  • Either Washer or Shell method can set up the volume of any solid of revolution.

  • Method selection depends on simplicity: one method may require a single integral while the other requires splitting into a sum of two or more integrals.

Arc Length of Plane Curves

• Arc Length Formula for Curves Function of $x$:

  • For smooth curve $y = f(x)$ on interval $a \le x \le b$:

  • \n\text{Length} = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx\n

• Arc Length Formula for Curves Function of $y$:

  • For smooth curve $x = g(y)$ on interval $c \le y \le d$:

  • \n\text{Length} = \int_{c}^{d} \sqrt{1 + (g'(y))^2} \, dy\n

• Algebraic Simplification Techniques for Radicands:

  • Evaluating arc length integrals often requires expanding and rewriting the radicand $1 + (f'(x))^2$ as a perfect square $[P(x)]^2$ so that \n\sqrt{[P(x)]^2} = P(x)\n

  • Requires showing complete algebraic work using binomial identities:

  • \n(a - b)^2 = a^2 - 2ab + b^2\n

  • \n(a + b)^2 = a^2 + 2ab + b^2\n

Area of Surfaces of Revolution

• Surface Area Revolving Curve $y = f(x)$ around the x-axis ($y = 0$):

  • Curve defined on $a \le x \le b$ with $f(x) \ge 0$:

  • \n\text{Surface area} = \int_{a}^{b} 2\pi f(x) \sqrt{1 + (f'(x))^2} \, dx\n

• Surface Area Revolving Curve $x = g(y)$ around the y-axis ($x = 0$):

  • Curve defined on $c \le y \le d$ with $g(y) \ge 0$:

  • \n\text{Surface area} = \int_{c}^{d} 2\pi g(y) \sqrt{1 + (g'(y))^2} \, dy\n

• General Surface Area Formula for Arbitrary Horizontal Axis $y = k$:

  • Formula:

  • \n\text{Surface area} = \int_{a}^{b} 2\pi r(x) \sqrt{1 + (f'(x))^2} \, dx\n

  • $r(x)$ represents the radius of revolution, defined as the perpendicular distance from point $(x, f(x))$ on the curve to the axis $y = k$:

  • If curve lies above axis $y = k$: $r(x) = f(x) - k$

  • If curve lies below axis $y = k$: $r(x) = k - f(x)$

  • Visual Representation:

  • 
    

  

• General Surface Area Formula for Arbitrary Vertical Axis:

  • Formulated by integration with respect to $y$ (or $x$) using radius $r$ measured horizontally from point $(g(y), y)$ to the vertical axis.