The Definition of Area as a Limit; Sigma Notation

The Definition of Area as a Limit & Sigma Notation

The concept of area as a limit is the foundation of definite integrals. It is based on approximating the area under a curve using rectangles, and then refining the approximation by taking the limit as the number of rectangles approaches infinity. Sigma ($\Sigma$) notation is used to express these sums concisely.

1. Approximating Area Using Rectangles (Riemann Sums)

To approximate the area under a curve $y = f(x)$ on the interval $[a, b]$:

1. Divide the interval into $n$ subintervals of equal width:

   $$\Delta x = \frac{b-a}{n}$$

2. Choose sample points $x_i^*$ in each subinterval.

3. Form rectangles using $f(x_i^*)$ as the height.

4. Compute the sum of the rectangle areas:

   $$A_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$$

5. Take the limit as $n \to \infty$ to get the exact area:

   $$A = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$$

This limit defines the definite integral:

2. Sigma Notation ($\Sigma$)

Sigma notation is a shorthand way of writing sums. It is written as:

where:

• $i$ is the index of summation.
• $a_i$ is the term being summed.
• $n$ is the number of terms.

For example:

3. Riemann Sums in Sigma Notation

Using sigma notation, the sum of areas of rectangles is:

• If $x_i^*$ is the left endpoint, it's called a left Riemann sum.
• If $x_i^*$ is the right endpoint, it's called a right Riemann sum.
• If $x_i^*$ is the midpoint, it's called a midpoint Riemann sum.

As $n \to \infty$, the sum approaches the exact area under the curve.

4. Example: Approximating an Integral Using Sigma Notation

Find the approximate area under $f(x) = x^2$ on $[0, 2]$ using 4 rectangles with right endpoints.

Step 1: Find $\Delta x$

Step 2: Identify Right Endpoints

Right endpoints:

Step 3: Compute Function Values

Step 4: Compute Approximate Area

Step 5: Exact Area Using Integration

As $n \to \infty$, the Riemann sum approaches the integral's value.

5. Importance of Area as a Limit

• Foundation of Definite Integrals
• Used in Physics (Work, displacement, fluid flow)
• Economics (Revenue, cost functions)
• Probability (Continuous probability distributions)

Understanding area as a limit through Riemann sums builds the intuition for definite integration.