5.1 Approximating Areas

Sigma notation simplifies complex sums, making it easier to work with series and sequences. It's a powerful tool for representing patterns and calculating totals, setting the stage for more advanced mathematical concepts.

Area estimation using rectangles and Riemann sums are key techniques for approximating definite integrals. These methods provide a visual and practical approach to understanding the fundamental theorem of calculus and its applications.

Sigma Notation and Summations

Summations with sigma notation

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• Sigma notation compactly represents the sum of a series of terms using the Greek letter $\Sigma$
• The index of summation (usually $i$ or $n$) is written below the $\Sigma$ symbol
• The starting and ending values of the index are written above and below the $\Sigma$, respectively ($\sum_{i=1}^{n}$)
• The general term of the series is written to the right of the $\Sigma$ ($\sum_{i=1}^{n} a_i$)
• To interpret a given summation, identify the starting and ending values of the index and determine the general term of the series
• Evaluate the summation by substituting the index values and simplifying ($\sum_{i=1}^{3} i^2 = 1^2 + 2^2 + 3^2 = 14$)

Approximating Areas

Area estimation using rectangles

• Approximating the area under a curve using rectangles is a fundamental concept in integral calculus
• Divide the interval $[a, b]$ into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$
• Construct rectangles on each subinterval with width $\Delta x$
  • The height of each rectangle can be determined by the function value at the left endpoint (left Riemann sum), right endpoint (right Riemann sum), or midpoint (midpoint Riemann sum) of the subinterval
• Calculate the area of each rectangle by multiplying its height by its width ($A_i = f(x_i^*) \Delta x$)
• Sum the areas of all the rectangles to approximate the total area under the curve ($A \approx \sum_{i=1}^{n} f(x_i^*) \Delta x$)
• As the number of subintervals $n$ increases, the approximation becomes more accurate ($\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_a^b f(x) dx$)

Riemann sums for definite integrals

• A Riemann sum is a method for approximating the definite integral of a function $f(x)$ over an interval $[a, b]$
• Divide the interval $[a, b]$ into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$
• Choose a sample point $x_i^*$ within each subinterval $[x_{i-1}, x_i]$ (left endpoint, right endpoint, or midpoint)
• Evaluate the function at each sample point to obtain $f(x_i^*)$
• The Riemann sum is given by $\sum_{i=1}^n f(x_i^*) \Delta x$
• The definite integral $\int_a^b f(x) dx$ is the limit of the Riemann sum as $n$ approaches infinity and $\Delta x$ approaches zero ($\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$)
• Riemann sums provide a way to approximate the value of a definite integral numerically ($\int_0^1 x^2 dx \approx \sum_{i=1}^{100} (\frac{i}{100})^2 \cdot \frac{1}{100}$)

Approximation techniques and limits

• The concept of area under a curve is fundamental to understanding definite integrals
• Approximation methods, such as Riemann sums, are used to estimate the area under a curve
• A partition of an interval [a,b] divides it into subintervals, which forms the basis for approximation techniques
• The limit process is crucial in refining approximations to obtain exact values of definite integrals