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Calculus II
Table of Contents
Unit 1 Overview: Integration
1.1 Approximating Areas
1.2 The Definite Integral
1.3 The Fundamental Theorem of Calculus
1.4 Integration Formulas and the Net Change Theorem
1.5 Substitution
1.6 Integrals Involving Exponential and Logarithmic Functions
1.7 Integrals Resulting in Inverse Trigonometric Functions

➗calculus ii review

1.1 Approximating Areas

Citation:

Sigma notation simplifies complex sums, making calculations more manageable. It's a powerful tool for representing sequences and series, allowing us to work with large datasets efficiently. This concept is crucial for understanding more advanced mathematical ideas.

Approximating areas under curves is a fundamental application of calculus. By using rectangular approximations and Riemann sums, we can estimate the area under a curve with increasing accuracy. This lays the groundwork for understanding definite integrals and their real-world applications.

Sigma Notation and Summation

Sigma notation for integer sums

  • Shorthand notation represents sum of sequence of numbers using Greek letter $\Sigma$
  • General form $\sum_{i=a}^{b} f(i)$ where:
    • $i$ index of summation (variable changes with each term)
    • $a$ lower limit of summation (starting value of $i$)
    • $b$ upper limit of summation (ending value of $i$)
    • $f(i)$ function or expression being summed ($i^2$, $2i+1$)
  • Properties of summation simplify calculations:
    • Constant multiple rule $\sum_{i=a}^{b} cf(i) = c\sum_{i=a}^{b} f(i)$ where $c$ constant ($\sum_{i=1}^{5} 3i = 3\sum_{i=1}^{5} i$)
    • Sum rule $\sum_{i=a}^{b} [f(i) + g(i)] = \sum_{i=a}^{b} f(i) + \sum_{i=a}^{b} g(i)$ ($\sum_{i=1}^{3} (i^2 + 2i) = \sum_{i=1}^{3} i^2 + \sum_{i=1}^{3} 2i$)
    • Difference rule $\sum_{i=a}^{b} [f(i) - g(i)] = \sum_{i=a}^{b} f(i) - \sum_{i=a}^{b} g(i)$ ($\sum_{i=1}^{4} (i^3 - i) = \sum_{i=1}^{4} i^3 - \sum_{i=1}^{4} i$)
  • Common summation formulas for integer sequences:
    • Sum of integers from 1 to $n$: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ ($\sum_{i=1}^{10} i = \frac{10(11)}{2} = 55$)
    • Sum of squares from 1 to $n$: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ ($\sum_{i=1}^{5} i^2 = \frac{5(6)(11)}{6} = 55$)
    • Sum of cubes from 1 to $n$: $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$ ($\sum_{i=1}^{4} i^3 = \left(\frac{4(5)}{2}\right)^2 = 100$)

Approximating Areas

Rectangular approximations for curve areas

  • Estimate area under a curve by dividing interval into subintervals and using rectangles to approximate area
  • Left Riemann sum uses left endpoint of each subinterval to determine rectangle height
    • Formula: $L_n = \sum_{i=1}^{n} f(x_{i-1})\Delta x$ where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$
    • Underestimates area for increasing functions, overestimates for decreasing functions
  • Right Riemann sum uses right endpoint of each subinterval to determine rectangle height
    • Formula: $R_n = \sum_{i=1}^{n} f(x_i)\Delta x$ where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$
    • Overestimates area for increasing functions, underestimates for decreasing functions
  • Midpoint Riemann sum uses midpoint of each subinterval to determine rectangle height
    • Formula: $M_n = \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right)\Delta x$ where $\Delta x = \frac{b-a}{n}$ and $x_i = a + i\Delta x$
    • Generally more accurate than left or right Riemann sums
  • Approximations become more accurate and converge to actual area as number of subintervals $n$ increases ($n=10$, $n=100$, $n=1000$)

Upper and Lower Sums

  • Upper sum: Uses maximum function value in each subinterval to create rectangles
    • Overestimates the area under the curve
  • Lower sum: Uses minimum function value in each subinterval to create rectangles
    • Underestimates the area under the curve
  • As the number of subintervals increases, upper and lower sums converge to the actual area

Riemann sums for definite integrals

  • Definite integral $\int_a^b f(x) dx$ represents exact area under curve $f(x)$ from $x=a$ to $x=b$
  • Riemann sums approximate definite integral value by summing rectangle areas
  • Limit of Riemann sums as number of subintervals approaches infinity equals definite integral:
    • $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^)\Delta x$ where $x_i^$ point in $i$-th subinterval and $\Delta x = \frac{b-a}{n}$
  • Steps to compute Riemann sum:
    1. Divide interval $[a, b]$ into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$
    2. Choose point $x_i^*$ in each subinterval (left endpoint, right endpoint, midpoint)
    3. Evaluate function $f(x_i^*)$ at each chosen point
    4. Multiply each function value by width $\Delta x$ to find rectangle area
    5. Sum rectangle areas using sigma notation ($\sum_{i=1}^{n} f(x_i^*)\Delta x$)
  • Riemann sum converges to definite integral as $n$ approaches infinity, providing more accurate area approximation ($\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\Delta x = \int_a^b f(x) dx$)
  • The partition of the interval $[a, b]$ into subintervals is crucial for creating Riemann sums

Key Terms to Review (30)

Area under the curve: The area under the curve represents the integral of a function over a given interval on a graph. It quantifies the accumulated value between the curve and the x-axis.
Definite integral: The definite integral of a function between two points provides the net area under the curve from one point to the other. It is represented by the integral symbol with upper and lower limits.
Dummy variable: A dummy variable is an arbitrary variable used to denote the variable of integration in a definite or indefinite integral. It has no effect on the value of the integral and can be replaced by any other symbol without changing the result.
Regular partition: A regular partition is a way of dividing an interval $[a, b]$ into subintervals of equal length. It is used to approximate areas under curves and in the computation of Riemann sums.
Summation notation: Summation notation, often represented by the Greek letter sigma ($\Sigma$), is a concise way to express the sum of a sequence of numbers. It is commonly used to approximate areas under curves and in various mathematical analyses.
Midpoint rule: The midpoint rule is a numerical integration technique used to approximate the definite integral of a function. It estimates the integral by taking the value of the function at the midpoint of each subinterval and multiplying it by the width of the subintervals.
Upper sum: The upper sum is an approximation of the area under a curve using the sum of the areas of rectangles that overestimate the region. Each rectangle's height is determined by the maximum function value over a subinterval.
Method of exhaustion: The method of exhaustion is an ancient mathematical technique used to find the area under a curve by inscribing and circumscribing polygons whose areas converge to the desired value. It involves taking the limit of the sum of the areas of these polygons as their number increases indefinitely.
Right-endpoint approximation: Right-endpoint approximation is a method for estimating the definite integral of a function. It uses the value of the function at the right endpoint of each subinterval to create rectangles whose areas are summed.
Index: An index is a variable used to denote the position of an element within a sequence or array, often used in summation notation. In integration, indices help identify terms in approximations like Riemann sums.
Left-endpoint approximation: Left-endpoint approximation is a method for estimating the area under a curve by summing the areas of rectangles whose heights are determined by the function value at the left endpoints of subintervals. It provides an estimate that can be used to approximate definite integrals.
Lower sum: The lower sum is an approximation of the area under a curve using the sum of the areas of inscribed rectangles. Each rectangle's height is determined by the minimum function value within each subinterval.
Sums and powers of integers: Sums and powers of integers are mathematical operations used to compute series of numbers raised to specific powers. These operations are essential in approximating areas under curves using techniques such as Riemann sums.
Midpoint Rule: The midpoint rule is a numerical integration method used to approximate the definite integral of a function over a given interval. It involves evaluating the function at the midpoint of the interval and multiplying the result by the width of the interval.
Upper Sum: The upper sum is a method used to approximate the area under a curve by dividing the interval into subintervals and constructing rectangles above the curve. The height of each rectangle is determined by the maximum value of the function within the corresponding subinterval, and the width of the rectangles is the length of the subinterval. As the number of subintervals increases, the upper sum provides a better approximation of the actual area under the curve.
Numerical Integration: Numerical integration is a mathematical technique used to approximate the definite integral of a function when an analytical solution is not readily available or practical to obtain. It involves breaking down the region under a curve into smaller segments and using numerical methods to estimate the area of those segments, ultimately providing an approximate value for the integral.
Riemann: Riemann is a fundamental concept in calculus that refers to the mathematical framework developed by the German mathematician Bernhard Riemann for defining and analyzing integrals. This concept is central to understanding the topics of approximating areas, the definite integral, integration formulas, and improper integrals in calculus.
Sigma Notation: Sigma notation, also known as summation notation, is a mathematical symbolism used to represent the sum of a series of terms or values. It is a concise way to express the addition of multiple quantities that follow a specific pattern or formula.
Subinterval: A subinterval is a smaller interval contained within a larger interval. It is a fundamental concept in calculus, particularly in the context of approximating areas under a curve using techniques like Riemann sums and Reimann integration.
Quadrature: Quadrature is the process of approximating the area under a curve using numerical integration techniques. It involves dividing the region into smaller segments and applying mathematical formulas to estimate the total area.
Simpson's Rule: Simpson's rule is a numerical integration method used to approximate the definite integral of a function over an interval. It is a more accurate alternative to the Riemann sum method, particularly for functions that can be well approximated by quadratic polynomials.
Antiderivative: An antiderivative, also known as a primitive function or indefinite integral, is a function whose derivative is the original function. It represents the accumulation or the reverse process of differentiation, allowing us to find the function that was differentiated to obtain a given derivative.
Left Endpoint: The left endpoint refers to the starting point or lower bound of an interval or range in the context of approximating areas. It is a crucial concept in numerical integration techniques, such as the Riemann sum, that are used to estimate the area under a curve.
Trapezoidal Rule: The trapezoidal rule is a numerical integration method used to approximate the definite integral of a function over a given interval. It is a simple and widely-used technique for estimating the area under a curve by dividing the interval into trapezoids and summing their areas.
Definite Integral: The definite integral represents the area under a curve on a graph over a specific interval. It is a fundamental concept in calculus that allows for the quantification of the accumulation of a quantity over a given range.
Simpson: Simpson's rule is a numerical method used to approximate the definite integral of a function. It is a type of quadrature formula that uses a parabolic interpolation to estimate the area under a curve, providing a more accurate approximation than the simpler Trapezoidal rule.
Lower Sum: The lower sum is a method used in calculus to approximate the area under a curve. It involves dividing the region into a finite number of vertical strips and calculating the sum of the areas of the rectangles formed by the lower bounds of each strip, providing a conservative estimate of the true area.
Area Under a Curve: The area under a curve represents the integral of a function over a specified interval, essentially calculating the accumulation of quantities represented by that function. This concept is crucial in understanding how to approximate and compute areas, connecting the graphical representation of functions with their corresponding numerical values. It also serves as a foundation for various mathematical applications, such as finding total distance traveled, work done, or other quantities derived from a function's rate of change.
Partition: A partition refers to the division of a certain interval into smaller sub-intervals, which is crucial for approximating areas under curves and ultimately leads to the concept of definite integrals. By breaking an interval into these smaller segments, it's possible to estimate the area more accurately using shapes like rectangles or trapezoids. This method of breaking things down helps to refine calculations and approaches the actual area as the number of partitions increases.
Right endpoint: A right endpoint refers to the upper boundary of an interval on the x-axis in the context of approximating areas under curves. This point is essential when calculating Riemann sums, specifically the right Riemann sum, which uses the value of the function at the right endpoint of each subinterval to estimate the area. This method highlights how choosing different endpoints can affect the accuracy of area approximations.