1.1 Approximating Areas - Calculus Volume 2

Learning Objectives

• 1.1.1 Use sigma (summation) notation to calculate sums and powers of integers.
• 1.1.2 Use the sum of rectangular areas to approximate the area under a curve.
• 1.1.3 Use Riemann sums to approximate area.

Introduction

• Archimedes used the method of exhaustion to calculate areas by filling irregular regions with shapes of known area to approximate total area.
• Techniques developed to approximate area between a curve defined by a function ( f(x) ) and the x-axis on a closed interval ([a,b]).
• Using smaller and smaller rectangles to get closer approximations.

Sigma (Summation) Notation

• Sigma notation is used to express long sums in a compact form.
• Example: Sum of integers from 1 to 20 is expressed as ( \sum_{i=1}^{20} i ).
• Properties of Sigma Notation:
  1. ( \sum_{i=1}^{n} c = nc )
  2. ( \sum_{i=1}^{n} c a_i = c \sum_{i=1}^{n} a_i )
  3. ( \sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i )
  4. ( \sum_{i=1}^{n} (a_i b_i) = \sum_{i=1}^{n} a_i \cdot \sum_{i=1}^{n} b_i )
  5. ( \sum_{i=1}^{n} a_i = \sum_{i=1}^{m} a_i + \sum_{i=m+1}^{n} a_i )_

Sums and Powers of Integers

• Sum of ( n ) integers: ( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} )
• Sum of consecutive integers squared: ( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} )
• Sum of consecutive integers cubed: ( \sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4} )_

Approximating Area Under a Curve

• Divide the interval ([a,b]) into ( n ) subintervals of equal width.
• Use Left-Endpoint Approximation and Right-Endpoint Approximation to estimate area.

Left-Endpoint Approximation

• Construct rectangles with height based on the function value at the left endpoint of the subinterval.
• Area approximation ( A \approx L_n = \sum_{i=1}^{n} f(x_{i-1}) \Delta x )

Right-Endpoint Approximation

• Construct rectangles with height based on the function value at the right endpoint of the subinterval.
• Area approximation ( A \approx R_n = \sum_{i=1}^{n} f(x_i) \Delta x )_

Forming Riemann Sums

• Evaluate function at any point within each subinterval ([x_{i-1}, x_i]).
• Riemann Sum: ( A_i = \sum_{i=1}^{n} f(x_i^) \Delta x )

Defining Area Under a Curve with Riemann Sums

• ( A = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x )
• Taking a limit of a sum computes the exact area.*

Lower and Upper Sums

• Lower sum: Choose points for minimum function value.
• Upper sum: Choose points for maximum function value.
• Practical for determining overestimates and underestimates.

Example Applications

• Examples provided demonstrate how to use sigma notation and approximations to compute specific areas and sums.
• Exercises and applications highlight practical uses and computational techniques.

Conclusion

• As ( n ) increases, approximations improve, converging to the exact area.
• Exercises provided for further practice and understanding of concepts and techniques discussed.