∫Calculus I Unit 5 – Integration

What's Integration All About?

• Integration calculates the area under a curve, the opposite of differentiation which finds the slope of a curve at a point
• Enables finding the accumulation of a quantity over an interval, such as the total distance traveled by an object
• Represented by the integral symbol $\int$, with the function being integrated called the integrand
• Definite integrals have specific start and end points (bounds), while indefinite integrals lack bounds and result in a function plus a constant C
• Fundamental Theorem of Calculus connects differentiation and integration, stating that integrating a function $f(x)$ over an interval $[a,b]$ is equivalent to evaluating an antiderivative $F(x)$ at the bounds: $\int_a^b f(x) dx = F(b) - F(a)$
• Integration by substitution is a key technique that simplifies complex integrands by introducing a new variable (u-substitution)
• Integrals have numerous real-world applications (area, volume, work, average value)

Key Concepts and Definitions

• Antiderivative: A function $F(x)$ whose derivative is the original function $f(x)$, satisfying $F'(x) = f(x)$
• Indefinite Integral: The set of all antiderivatives of a function, denoted as $\int f(x) dx = F(x) + C$, where C is an arbitrary constant
• Definite Integral: The limit of a Riemann sum as the number of subdivisions approaches infinity, represented as $\int_a^b f(x) dx$, where a and b are the lower and upper bounds
• Riemann Sum: An approximation of the area under a curve by dividing the interval into subintervals and summing the areas of rectangles
• Fundamental Theorem of Calculus (Part 1): If $F(x)$ is an antiderivative of $f(x)$ on $[a,b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
• Fundamental Theorem of Calculus (Part 2): If $f(x)$ is continuous on $[a,b]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
• Integration by Substitution: A technique for simplifying integrals by introducing a new variable, typically denoted as u, to transform the integrand

Integration Techniques

• U-substitution transforms the integral of a composite function into a simpler form by setting $u = g(x)$, $du = g'(x)dx$, and rewriting the integral in terms of u
• Integration by parts is used when the integrand is a product of functions, utilizing the formula $\int u dv = uv - \int v du$
  • Commonly applied to integrals involving products of polynomials and trigonometric, exponential, or logarithmic functions
• Trigonometric substitution simplifies integrals containing $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$ by introducing a trigonometric function (sine, cosine, or tangent)
• Partial fraction decomposition breaks down rational functions into simpler fractions that can be integrated separately
  • Useful for integrands that are ratios of polynomials
• Integration by tables involves recognizing common integral forms and their corresponding antiderivatives from a reference table
• Numerical integration methods (Trapezoidal Rule, Simpson's Rule) approximate definite integrals using discrete points when an exact antiderivative is difficult to find or the function is given as a set of data points

Applications of Integration

• Area between curves: Calculate the area enclosed by two or more functions by subtracting their integrals over the desired interval
• Volumes of solids: Find the volume of a solid formed by rotating a region around an axis using the disk method ($\pi \int_a^b [f(x)]^2 dx$) or shell method ($2\pi \int_a^b x f(x) dx$)
• Arc length: Determine the length of a curve between two points using the formula $\int_a^b \sqrt{1 + [f'(x)]^2} dx$
• Surface area: Calculate the surface area of a solid formed by rotating a curve around an axis with $2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} dx$
• Work: Compute the work done by a variable force $F(x)$ over a distance using $\int_a^b F(x) dx$
• Average value of a function: Find the average value of $f(x)$ over $[a,b]$ with $\frac{1}{b-a} \int_a^b f(x) dx$
• Center of mass: Determine the center of mass of a thin rod or lamina using integration

Common Mistakes and How to Avoid Them

• Forgetting to add the constant of integration (C) when finding an indefinite integral
  • Always include +C unless otherwise specified
• Incorrectly applying the Fundamental Theorem of Calculus by not evaluating the antiderivative at the bounds
  • Remember to substitute the upper and lower bounds into the antiderivative and subtract: $F(b) - F(a)$
• Mishandling u-substitution by inconsistently substituting or forgetting to change the differential (dx)
  • Ensure that du is properly expressed in terms of dx, and substitute both u and du consistently
• Improperly choosing the integration technique for a given problem
  • Identify key characteristics of the integrand to select the most appropriate method (u-substitution, integration by parts, trigonometric substitution, partial fractions)
• Incorrectly setting up or evaluating definite integrals
  • Pay attention to the order of the bounds and the sign of the result when the lower bound is greater than the upper bound
• Misinterpreting or misapplying integration in word problems
  • Carefully read the problem statement, identify the relevant variables and functions, and set up the integral accordingly
• Algebraic or arithmetic errors in simplification and calculation
  • Double-check your work, simplify expressions step-by-step, and use a calculator for accurate arithmetic

Practice Problems and Solutions

1. Evaluate $\int (3x^2 + 2x - 1) dx$

   • Solution: $\int (3x^2 + 2x - 1) dx = x^3 + x^2 - x + C$

2. Find $\int_0^1 (4x^3 - 2x + 3) dx$

   • Solution: $\int_0^1 (4x^3 - 2x + 3) dx = (x^4 - x^2 + 3x) \bigg|_0^1 = (1 - 1 + 3) - (0 - 0 + 0) = 3$

3. Evaluate $\int x\sqrt{1+x^2} dx$ using u-substitution

   • Solution: Let $u = 1+x^2$, then $du = 2x dx$ or $\frac{1}{2} du = x dx$
     • $\int x\sqrt{1+x^2} dx = \frac{1}{2} \int \sqrt{u} du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{3} (1+x^2)^{3/2} + C$

4. Calculate the area between the curves $y = x^2$ and $y = x + 2$ over the interval $[0, 2]$

   • Solution: Area $= \int_0^2 (x + 2) dx - \int_0^2 x^2 dx = \left(\frac{1}{2}x^2 + 2x\right) \bigg|_0^2 - \left(\frac{1}{3}x^3\right) \bigg|_0^2 = (4 + 4) - (\frac{8}{3}) = \frac{10}{3}$

5. Find the volume of the solid formed by rotating the region bounded by $y = \sqrt{x}$, $y = 0$, $x = 1$, and $x = 4$ about the x-axis

   • Solution: Using the disk method, $V = \pi \int_1^4 (\sqrt{x})^2 dx = \pi \int_1^4 x dx = \pi \left(\frac{1}{2}x^2\right) \bigg|_1^4 = \frac{\pi}{2} (16 - 1) = \frac{15\pi}{2}$

Real-World Examples

• Calculating the work done by a variable force (spring, fluid pressure) over a distance
  • Example: A spring with force $F(x) = kx$ compressed from 0 to 0.2 meters with $k = 500 N/m$ does $W = \int_0^{0.2} 500x dx = 10$ joules of work
• Determining the volume of irregular objects (vases, containers) by rotation or cross-sections
  • Example: The volume of a vase formed by rotating $y = x^2$ around the y-axis from 0 to 3 is $V = \pi \int_0^3 x^4 dx = \frac{243\pi}{5}$ cubic units
• Finding the center of mass or moment of inertia of non-uniform objects
  • Example: The center of mass of a thin rod with linear density $\rho(x) = x$ from 0 to 1 is $\bar{x} = \frac{\int_0^1 x^2 dx}{\int_0^1 x dx} = \frac{1/3}{1/2} = \frac{2}{3}$
• Calculating the average value of a function over an interval (mean value theorem)
  • Example: The average velocity of an object with velocity $v(t) = t^2$ from $t = 0$ to $t = 2$ seconds is $\frac{1}{2-0} \int_0^2 t^2 dt = \frac{8}{3}$ units per second
• Determining the area or volume of irregular shapes in engineering and design
  • Example: The area of a plot of land bounded by a river and a road can be approximated using integration and surveying data

Connecting Integration to Other Math Topics

• Integration is the inverse operation of differentiation, with the Fundamental Theorem of Calculus linking the two concepts
• Integrals are used in probability theory to calculate expected values and probabilities of continuous random variables
  • Example: The probability density function of a normal distribution is integrated to find probabilities
• Integration is essential in solving differential equations, which model various phenomena in physics, engineering, and economics
  • Example: The solution to a first-order linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ involves integrating the integrating factor $e^{\int P(x) dx}$
• Fourier analysis uses integration to represent functions as sums of trigonometric functions (Fourier series) or integrals of complex exponentials (Fourier transforms)
  • Example: The Fourier coefficients of a periodic function $f(x)$ are calculated using $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$ and $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$
• Vector calculus extends integration to multiple dimensions, with concepts like line integrals, surface integrals, and volume integrals
  • Example: The flux of a vector field $\mathbf{F}$ through a surface $S$ is calculated using the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$
• Integration is used in statistics to find moments, cumulative distribution functions, and marginal and conditional distributions of multivariate random variables
  • Example: The expected value of a continuous random variable $X$ with probability density function $f(x)$ is $E[X] = \int_{-\infty}^{\infty} x f(x) dx$