91Ó°ÊÓ

The geometric interpretation of a definite integral helps us visualize what the integral of a function, particularly a constant function, represents. For a constant function \( f(x) = C \), imagine it as a straight horizontal line on a graph where every \( y \)-value is equal to \( C \). Now, consider the interval \([a, b]\) on the \( x \)-axis.
The area under this line (from \( a \) to \( b \)) forms a rectangle. The height of this rectangle is \( C \), because that is the value of the function throughout, and its width is \( b-a \), which is the length of the interval on the \( x \)-axis. To find the area of the rectangle, we use the formula for the area of a rectangle:

• Area = height \( \times \) width

Substituting, we get:

• Area = \( C \times (b-a) \)

This area is exactly the value of the definite integral \( \int_{a}^{b} f(x) \, dx \). Thus, the geometric interpretation confirms that the integral of the constant function over \([a, b]\) is equal to the area of this rectangle, \( C(b-a) \). This visualization makes it easier to grasp the concept of integrating a constant function.

A Riemann sum is a method for approximating the total area under a curve on a graph, which is a way of estimating the value of a definite integral. In the context of the constant function \( f(x) = C \), it serves as a concrete method to confirm the result found using the geometric interpretation.
To construct a Riemann sum, the interval \([a, b]\) is divided into \( n \) equally spaced subintervals. The width of each subinterval, called \( \Delta x \), is given by:

• \( \Delta x = \frac{b-a}{n} \)

For each subinterval, we select a point \( x_i^* \), often the right endpoint, and evaluate the function there. Because \( f(x) = C \), the function value does not change, and each function evaluation is simply \( C \). Therefore, every term in the Riemann sum is calculated as
\( C \times \Delta x \).

• The complete Riemann sum is \( \sum_{i=1}^{n} C \cdot \Delta x = C \cdot \sum_{i=1}^{n} \Delta x \)

Given that \( \sum_{i=1}^{n} \Delta x = n \times \Delta x \) and \( \Delta x = \frac{b-a}{n} \), we derive:

• \( C \cdot n \cdot \frac{b-a}{n} = C(b-a) \)

This reafirms the area under the curve (the definite integral) as \( C(b-a) \). So, Riemann sums help us approximate the integral, eventually revealing the exact value as \( n \) approaches infinity.

The definition of a definite integral formalizes the process of finding the accumulated area under a curve over an interval \([a, b]\). It's a mathematical concept that captures the essence of summing infinitely many infinitesimally small quantities, represented by area segments.
For a constant function \( f(x) = C \), the definite integral \( \int_{a}^{b} f(x) \, dx \) is directly related to the limit of its Riemann sums as the number of subintervals \( n \) goes to infinity. This is precisely defined as:

• \( \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \cdot \Delta x \)

In simpler terms, as \( n \) increases, each subinterval becomes narrower, and our approximation becomes more accurate, approaching the exact area under the curve.
For our constant function, every Riemann sum evaluates to \( C \cdot (b-a) \), and "taking the limit" is trivial because the sum does not change with \( n \). Thus, each calculation results in:

• \( \int_{a}^{b} f(x) \, dx = C(b-a) \)

This definition not only validates the calculations obtained using geometric and Riemann sum approaches, but also unifies these methods under a rigorous mathematical framework.