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