Lecture on Finding Area Under a Curve with Riemann Sums

Key Concepts

• Riemann Sums: Method to approximate the area under a curve.
• Definite Integral: Exact calculation of the area under a curve from point (a) to (b).
• Endpoints: Used in Riemann sums to calculate the area with left, right, and midpoint methods.

Function Example

• Consider (f(x) = x^2 + 1).
  • Parabola starting at 1 and opens upward.
  • Calculate area from (x = 0) to (x = 2).

Definite Integral Calculation

1. Integrate (f(x) = x^2 + 1) to find the antiderivative.
2. Evaluate from 0 to 2:
   • (\int_0^2 (x^2 + 1) , dx = \left[\frac{x^3}{3} + x\right]0^2 = \frac{8}{3} + 2 = \frac{14}{3} \approx 4.67)

Riemann Sums

• Delta x (width of rectangles): (\Delta x = \frac{b-a}{n})
• Left Endpoint Approximation:
  • Use points (0, 0.5, 1, 1.5).
  • Result: 3.75 (under-approximation).
• Right Endpoint Approximation:
  • Use points (0.5, 1, 1.5, 2).
  • Result: 5.75 (over-approximation).
• Midpoint Rule:
  • Use points (0.25, 0.75, 1.25, 1.75).
  • Result: 4.625 (most accurate of three).

Observations

• Increasing (n) (number of rectangles) increases approximation accuracy.
• Averaging left and right Riemann sums gives a closer approximation.

Visual Representations

• Graphing Rectangles:
  • Left endpoint: rectangles below the curve.
  • Right endpoint: rectangles above the curve.
  • Midpoint: balances areas above and below the curve.

Example with Different Function

• Function: (f(x) = x^3), ([0, 4])
  • Calculate definite integral: (\int_0^4 x^3 , dx).
  • Area using left: 36, and right: 100.
  • Midpoint provides closer estimate.

Decreasing Functions

• Example: (f(x) = \frac{1}{x}), ([1, 2])
  • Actual area: (\approx 0.6931).
  • Left endpoint: over-approximation.
  • Right endpoint: under-approximation.

Sigma Notation and Limits

• Defining Definite Integral with Limit: [ \text{Area} = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x ]
  • (\Delta x = \frac{b-a}{n})
  • Right Endpoint Formula: (x_i = a + i\Delta x)
  • Left Endpoint Formula: (x_i = a + (i-1)\Delta x)

Equations

• Sum of constant (c): (cn)
• Sum of (i): (\frac{n(n+1)}{2})
• Sum of (i^2): (\frac{n(n+1)(2n+1)}{6})
• Sum of (i^3): (\left(\frac{n(n+1)}{2}\right)^2)

Final Example

• Function: (f(x) = 5x - 2)
  • Interval: ([1, 4])
  • Definite integral: (\int_1^4 (5x - 2) , dx) gives (\frac{63}{2}).

Conclusion

• Using different Riemann sum methods and exact integrals provides a comprehensive understanding of approximating area under curves.