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.

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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:

$$A = \int_{a}^{b} f(x) \, dx$$

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2. Sigma Notation ($\Sigma$)

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

$$\sum_{i=1}^{n} a_i$$

where:

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

For example:

$$\sum_{i=1}^{4} i = 1 + 2 + 3 + 4 = 10$$

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3. Riemann Sums in Sigma Notation

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

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

• 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.

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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 $\Delta x$

$$\Delta x = \frac{b-a}{n} = \frac{2-0}{4} = 0.5$$

Step 2: Identify Right Endpoints

Right endpoints:

$$x_1^* = 0.5, \quad x_2^* = 1, \quad x_3^* = 1.5, \quad x_4^* = 2$$

Step 3: Compute Function Values

$$f(0.5) = (0.5)^2 = 0.25, \quad f(1) = 1, \quad f(1.5) = 2.25, \quad f(2) = 4$$

Step 4: Compute Approximate Area

$$A_4 = \sum_{i=1}^{4} f(x_i^*) \Delta x$$ $$= (0.25 + 1 + 2.25 + 4) \times 0.5$$ $$= 3.75$$

Step 5: Exact Area Using Integration

$$\int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{8}{3} \approx 2.67$$

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

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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.