Calculus II: Integration and Its Applications – Study Notes (Weeks 1–4)

Integration and Its Applications

Indefinite Integrals

The indefinite integral represents the family of all antiderivatives of a function. It is a fundamental concept in calculus, used to reverse the process of differentiation.

• Definition: The indefinite integral of a function f(x) is written as $\int f(x)\,dx$ and represents all functions whose derivative is f(x).

• General Form: $\int f(x)\,dx = F(x) + C$, where F(x) is any antiderivative of f(x) and C is the constant of integration.

• Example: $\int x^2\,dx = \frac{1}{3}x^3 + C$

Substitution Rule

The substitution rule (also known as u-substitution) is a technique for evaluating integrals by making a substitution to simplify the integrand.

• Method: Let $u = g(x)$, then $\frac{du}{dx} = g'(x)$, so $dx = \frac{du}{g'(x)}$.

• Formula: $\int f(g(x))g'(x)\,dx = \int f(u)\,du$

• Example: $\int 2x\cos(x^2)\,dx$ Let $u = x^2$, $du = 2x\,dx$ So, $\int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C$

Regions Between Curves

To find the area between two curves, integrate the difference of their functions over the interval where they intersect.

• Formula: $A = \int_a^b [f(x) - g(x)]\,dx$, where $f(x) \geq g(x)$ on $[a, b]$.

• Example: Area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$: $A = \int_0^1 (x - x^2)\,dx = \left[\frac{1}{2}x^2 - \frac{1}{3}x^3\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Applications of Definite Integrals

Volume by Slicing

The method of slicing finds the volume of a solid by integrating the area of cross-sections perpendicular to an axis.

• Formula: $V = \int_a^b A(x)\,dx$, where $A(x)$ is the area of the cross-section at $x$.

• Example: Volume of a solid with square cross-sections of side $x$ from $x = 0$ to $x = 2$: $V = \int_0^2 x^2\,dx = \left[\frac{1}{3}x^3\right]_0^2 = \frac{8}{3}$

Volume by Shells

The cylindrical shells method is used to find the volume of solids of revolution by integrating the lateral surface area of cylindrical shells.

• Formula: $V = 2\pi \int_a^b r(x)h(x)\,dx$, where $r(x)$ is the radius and $h(x)$ is the height of the shell.

• Example: Volume generated by revolving $y = x$ from $x = 0$ to $x = 1$ about the y-axis: $V = 2\pi \int_0^1 x(x)\,dx = 2\pi \int_0^1 x^2\,dx = 2\pi \left[\frac{1}{3}x^3\right]_0^1 = \frac{2\pi}{3}$

Arc Length

The arc length of a curve $y = f(x)$ from $x = a$ to $x = b$ is found by integrating the square root of $1 + (f'(x))^2$.

• Formula: $L = \int_a^b \sqrt{1 + (f'(x))^2}\,dx$

• Example: Arc length of $y = x^2$ from $x = 0$ to $x = 1$: $L = \int_0^1 \sqrt{1 + (2x)^2}\,dx = \int_0^1 \sqrt{1 + 4x^2}\,dx$

Surface Area

The surface area of a solid of revolution is found by integrating the circumference of the revolving curve times the arc length differential.

• Formula (about x-axis): $S = 2\pi \int_a^b f(x)\sqrt{1 + (f'(x))^2}\,dx$

• Example: Surface area of $y = x$ from $x = 0$ to $x = 1$ revolved about the x-axis: $S = 2\pi \int_0^1 x\sqrt{1 + 1^2}\,dx = 2\pi \int_0^1 x\sqrt{2}\,dx = 2\pi \sqrt{2}\left[\frac{1}{2}x^2\right]_0^1 = \pi \sqrt{2}$

Work

Work done by a variable force is calculated using definite integrals.

• Formula: $W = \int_a^b F(x)\,dx$, where $F(x)$ is the force as a function of position.

• Example: Work to stretch a spring from $x = 0$ to $x = 2$ if $F(x) = 3x$: $W = \int_0^2 3x\,dx = \left[\frac{3}{2}x^2\right]_0^2 = 6$

Logarithmic, Exponential, and Hyperbolic Functions

Logarithm and Exponential Functions

Exponential and logarithmic functions are essential in calculus, especially for modeling growth and decay.

• Derivative of Exponential: $\frac{d}{dx}e^{x} = e^{x}$

• Integral of Exponential: $\int e^{x}\,dx = e^{x} + C$

• Derivative of Logarithm: $\frac{d}{dx}\ln(x) = \frac{1}{x}$

• Integral of Logarithm: $\int \ln(x)\,dx = x\ln(x) - x + C$

• Example: $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$

Exponential Growth and Decay

Many natural processes are modeled by exponential growth or decay, such as population growth or radioactive decay.

• General Solution: $y(t) = y_0 e^{kt}$, where $k > 0$ for growth, $k < 0$ for decay.

• Example (Decay): Carbon-14 dating uses $N(t) = N_0 e^{-kt}$ to model radioactive decay.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions, defined using exponential functions.

• Definitions:

  • $\sinh(x) = \frac{e^{x} - e^{-x}}{2}$

  • $\cosh(x) = \frac{e^{x} + e^{-x}}{2}$

• Derivatives:

  • $\frac{d}{dx}\sinh(x) = \cosh(x)$

  • $\frac{d}{dx}\cosh(x) = \sinh(x)$

• Integrals:

  • $\int \sinh(x)\,dx = \cosh(x) + C$

  • $\int \cosh(x)\,dx = \sinh(x) + C$

Techniques of Integration

Integration by Parts

Integration by parts is based on the product rule for differentiation and is used to integrate products of functions.

• Formula: $\int u\,dv = uv - \int v\,du$

• Example: $\int x e^{x}\,dx$ Let $u = x$, $dv = e^{x}\,dx$ Then $du = dx$, $v = e^{x}$ So, $\int x e^{x}\,dx = x e^{x} - \int e^{x}\,dx = x e^{x} - e^{x} + C$

Integral Tables

Integral tables provide a list of common integrals for quick reference, especially for complex functions.

• Usage: Match the integrand to a form in the table and apply the corresponding formula.

• Example: $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$

Numerical Integration

When an integral cannot be evaluated analytically, numerical methods such as the Trapezoidal Rule and Simpson's Rule are used to approximate its value.

• Trapezoidal Rule: $\int_a^b f(x)\,dx \approx \frac{b-a}{2n}\left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]$

• Simpson's Rule: $\int_a^b f(x)\,dx \approx \frac{b-a}{3n}\left[f(x_0) + 4\sum_{\text{odd }i} f(x_i) + 2\sum_{\text{even }i} f(x_i) + f(x_n)\right]$

• Error Bounds: Both methods have error estimates that depend on the function's derivatives and the number of subintervals.

Comparison of Numerical Integration Methods

Method	Formula	Accuracy	When to Use
	Trapezoidal Rule	$\frac{b-a}{2n}\left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]$	Moderate	Simple, quick estimates
Simpson's Rule	$\frac{b-a}{3n}\left[f(x_0) + 4\sum_{\text{odd }i} f(x_i) + 2\sum_{\text{even }i} f(x_i) + f(x_n)\right]$	Higher		When higherWeek 1 Integration Indefinite Integral Substitution Rule Regions Between Curves Calculus Early Transcendentals Chapter 5, Section 5 Chapter 6, Sections 1-2 Learning Resources from the Instructor Week 1 Instructor Summary Geogebra Interactive Area Between Curves Videos Substitution Method Examples Substitutions w/ Definite Integrals Areas Between Curves Ex. 1 Areas Between Curves Ex. 2 Yellowdig Participation Introduction Pearson My Lab Math (MML) Week 1 Homework Week 1 Quiz , Week 2 Applications of Definite Integrals Volume by Slicing Volume by Shells Arc Length Surface Area Work Calculus Early Transcendentals Chapter 6, Sections 3-7 Learning Resources from the Instructor Week 2 Instructor Summary Geogebra Interactive Visualizing Rotation about x-axis Visualizing Rotation about y-axis Videos Volume by Cross-Sectional Area Volumes of Solids of Revolution Arc Length Calculation Surface Area Calculation Calculating Work Yellowdig Participation Pearson My Lab Math (MML) Week 2 Homework Week 2 Quiz Week 3 Logarithmic, Exponential, and Hyperbolic Functions Logarithm and Exponential Functions Exponential Growth/Decay Hyperbolic Functions Calculus Early Transcendentals Chapter 7, Sections 1-3 Learning Resources from the Instructor Week 3 Instructor Summary Videos Integration w/Exponential and Logarithmic Functions Carbon-14 Dating Derivatives and Integrals of Hyperbolic Functions Yellowdig Participation Pearson My Lab Math (MML) Week 3 Homework Week 3 Quiz Week 4 Techniques of Integration Integration by Parts Integral Tables Numerical Integration Calculus Early Transcendentals Chapter 8, Sections 1,2,7, and 8 Learning Resources from the Instructor Week 4 Instructor Summary Geogebra Interactive Trapezoidal Rule and Simpson's Rule Interactivity Videos Integration by Parts Integrals Using Tables Error Bounds on Trapezoidal and Simpson's Rules Yellowdig Participation Pearson My Lab Math (MML) Week 4 Homework Midterm Examaccuracy is needed

Additional info: These notes are based on a course outline referencing "Calculus: Early Transcendentals" and cover Weeks 1–4, including integration, applications, and advanced techniques. For more detailed examples and practice, refer to the textbook and assigned homework.