Lecture Notes: Numerical Integration - The Trapezoidal Rule

Overview

• Purpose: Estimate the value of a definite integral, representing the area under a curve.
• Example Function: ( y = x^2 ), estimating the area from 0 to 10 using the trapezoidal rule.

Key Concepts

Delta ( x )

• Formula: ( \Delta x = \frac{b - a}{n} )
• Example Calculation:
  • ( a = 0 ), ( b = 10 ), ( n = 5 )
  • ( \Delta x = \frac{10 - 0}{5} = 2 )

Points on the Number Line

• Left Endpoint Rule: Use 5 out of 6 points: 0, 2, 4, 6, 8
• Right Endpoint Rule: Use 5 out of 6 points: 2, 4, 6, 8, 10
• Midpoint Rule: Use midpoints: 1, 3, 5, 7, 9
• Trapezoidal Rule: Use all points: 0, 2, 4, 6, 8, 10

Trapezoidal Rule Formula

• Formula: ( \text{Area} = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n) \right) )
• Note:
  • First and last ( y ) values are not multiplied by 2.
  • Middle ( y ) values are multiplied by 2._

Example Calculation

• Function: ( f(x) = x^2 )
• Points: 0, 2, 4, 6, 8, 10
• Values:
  • ( f(0) = 0^2 = 0 )
  • ( f(2) = 2^2 = 4 )
  • ( f(4) = 4^2 = 16 )
  • ( f(6) = 6^2 = 36 )
  • ( f(8) = 8^2 = 64 )
  • ( f(10) = 10^2 = 100 )
• Calculation:
  • ( \frac{2}{2} ) ( (0 + 2 \times 4 + 2 \times 16 + 2 \times 36 + 2 \times 64 + 100) = 340 )
• Comparison: Exact integral ( \int_0^{10} x^2 , dx = \frac{1000}{3} = 333.3 )

Application: Water Accumulation Problem

• Scenario: Estimate water accumulation in a tank over an hour with given rates every 10 minutes.
• Data Points: 7 points, 6 intervals
• Delta X Calculation:
  • Interval: 0 to 60 minutes
  • ( \Delta x = \frac{60 - 0}{6} = 10 \text{ minutes} )
• Trapezoidal Rule Application:
  • Use rates: 3.8, 4.5, 6.2, 7, 7.5, 6.9, 6.2
  • Calculation:
    • ( 5 \times (3.8 + 2 \times 4.5 + 2 \times 6.2 + 2 \times 7 + 2 \times 7.5 + 2 \times 6.9 + 6.2) = 371 \text{ gallons} )

Conclusion

• The Trapezoidal Rule provides a good approximation for evaluating definite integrals and estimating areas under curves or accumulated quantities.