Calculus II: Integration and Its Applications – Weeks 1-4 Study Guide

Integration and Its Applications

Indefinite Integrals

The indefinite integral represents the family of all antiderivatives of a function. It is written as:

• Definition: The indefinite integral of a function f(x) is $\int f(x)\,dx = F(x) + C$, where F(x) is any antiderivative of f(x) and C is the constant of integration.

• Properties: Linearity, sum/difference, and constant multiple rules apply.

• Example: $\int 3x^2\,dx = x^3 + C$

Substitution Rule

The substitution method (also called u-substitution) simplifies integration by changing variables.

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

• Steps:

  1. Let u be a function inside the integrand.

  2. Compute $du = g'(x)dx$.

  3. Rewrite the integral in terms of u and integrate.

  4. Substitute back in terms of x.

• 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: $\text{Area} = \int_a^b [f(x) - g(x)]\,dx$, where f(x) is the upper function and g(x) is the lower function.

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

Applications of Definite Integrals

Volume by Slicing

Volumes of solids can be found 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: For 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 shell method is useful for finding volumes of solids of revolution, especially when integrating parallel to the axis of rotation is easier.

• Formula: $V = 2\pi \int_a^b (\text{radius})(\text{height})\,dx$

• Example: Rotate $y = x$ 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 length of a curve between two points can be found using the arc length formula.

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

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

Surface Area

The surface area of a solid of revolution can be found by integrating the circumference of the revolving curve.

• 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 over a distance can be calculated using integration.

• Formula: $W = \int_a^b F(x)\,dx$

• Example: Lifting a rope with linear weight density $\lambda$ over a height $h$: $W = \int_0^h \lambda (h - x)\,dx$

Logarithmic, Exponential, and Hyperbolic Functions

Logarithm and Exponential Functions

Exponential and logarithmic functions are fundamental 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 (Exponential Growth): $\frac{dy}{dt} = ky$ has solution $y = y_0 e^{kt}$

Exponential Growth and Decay

Many natural processes follow exponential growth or decay, such as population growth or radioactive decay.

• General Solution: $y(t) = y_0 e^{kt}$

• Half-life Formula: $t_{1/2} = \frac{\ln 2}{|k|}$

• Example (Carbon-14 Dating): Used to determine the age of artifacts by measuring remaining $^{14}C$.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but based on hyperbolas.

• Definitions:

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

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

  • $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$

• 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

This technique is based on the product rule for differentiation and is useful for integrating products of functions.

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

• Choosing u and dv: Use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to choose u.

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

Integral Tables

Integral tables provide formulas for common integrals, useful for quickly finding antiderivatives.

• Usage: Match the integrand to a formula in the table and apply it, possibly after substitution.

• 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 approximate its value.

• Trapezoidal Rule: Approximates area under a curve using trapezoids. $\int_a^b f(x)\,dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n) \right]$

• Simpson's Rule: Uses parabolic arcs for better accuracy. $\int_a^b f(x)\,dx \approx \frac{b-a}{3n} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n) \right]$

• Error Bounds: Both methods have error estimates based on the function's derivatives.

Summary Table: Integration Techniques and Applications

Topic	Main Formula	Example
Indefinite Integral	$\int f(x)\,dx = F(x) + C$	$\int x^2\,dx = \frac{1}{3}x^3 + C$
Substitution	$\int f(g(x))g'(x)\,dx = \int f(u)\,du$	$\int 2x\cos(x^2)\,dx$
Area Between Curves	$\int_a^b [f(x) - g(x)]\,dx$	$\int_0^1 (x - x^2)\,dx$
Volume by Slicing	$\int_a^b A(x)\,dx$	Square cross-section: $\int_0^2 x^2\,dx$
Volume by Shells	$2\pi \int_a^b (\text{radius})(\text{height})\,dx$	$2\pi \int_0^1 x^2\,dx$
Arc Length	$\int_a^b \sqrt{1 + [f'(x)]^2}\,dx$	$\int_0^1 \sqrt{1 + 4x^2}\,dx$
Surface Area	$2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2}\,dx$	$2\pi \int_0^1 x\sqrt{2}\,dx$
Work	$\int_a^b F(x)\,dx$	$\int_0^h \lambda (h - x)\,dx$
Integration by Parts	$\int u\,dv = uv - \int v\,du$	$\int x e^{x}\,dx$
Numerical Integration	Trapezoidal/Simpson's Rule	See above formulas

Additional info: This guide is based on a course outline referencing "Calculus: Early Transcendentals" and covers core topics from Chapters 5-8, including both conceptual and computational aspects of integration and its applications.