Question

Suppose that \(f\) is continuous on \([a, b]\) and \(P=\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) is a partition of \([a, b]\). Show that there is a Riemann sum of \(f\) over \(P\) that equals \(\int_{a}^{b} f(x) d x\).

Short answer

Question: Prove that there exists a Riemann sum of a continuous function \(f\) over a partition \(P\) of the interval \([a, b]\) that is equal to the definite integral of \(f\) in the interval \([a, b]\). Answer: By applying the Intermediate Value Theorem and the properties of definite integrals, we can show that \(\sum_{i=0}^{n-1} f(\xi_i)\Delta x_i = \int_{a}^{b} f(x) dx\), which proves that there exists a Riemann sum of the continuous function \(f\) over the partition \(P\) that equals the definite integral of \(f\) in the interval \([a, b]\).

Step-by-step solution

Defining a Riemann sum

A Riemann sum is a method for approximating the definite integral of a function \(f\) over an interval \([a, b]\). It is defined as: $$\sum_{i=0}^{n-1} f(\xi_i)\Delta x_{i}$$ where \(P=\left\\{x_{0}, x_{1},..., x_{n}\right\\}\) is a partition of \([a, b]\), \(\Delta x_{i} = x_{i+1} - x_{i}\) is the length of each subinterval, and \(\xi_i\) is a sample point in the interval \([x_{i}, x_{i+1}]\).

Applying properties of continuous functions

Since \(f\) is continuous on \([a, b]\), by the Intermediate Value Theorem, for any subinterval \([x_i, x_{i+1}]\), there exists a point \(\xi_i \in [x_i, x_{i+1}]\) such that: $$f(\xi_i) = \frac{\int_{x_i}^{x_{i+1}}f(x) dx}{\Delta x_i}$$

Manipulate to find the sum of the integral

Multiply both sides of the equation in step 2 by \(\Delta x_i\): $$f(\xi_i)\Delta x_i = \int_{x_i}^{x_{i+1}}f(x) dx$$ Sum the equation over all subintervals: $$\sum_{i=0}^{n-1} f(\xi_i)\Delta x_i = \sum_{i=0}^{n-1} \int_{x_i}^{x_{i+1}}f(x) dx$$

Use the properties of definite integrals

Applying the properties of definite integrals, we can sum the right side of the equation in step 3: $$\sum_{i=0}^{n-1} \int_{x_i}^{x_{i+1}}f(x) dx = \int_{a}^{b} f(x) dx$$

Conclusion

From steps 3 and 4, we have: $$\sum_{i=0}^{n-1} f(\xi_i)\Delta x_i = \int_{a}^{b} f(x) dx$$ Thus, there is a Riemann sum of \(f\) over the partition \(P\) that equals the definite integral \(\int_{a}^{b} f(x) dx\).

Key concepts

Definite Integral

The definite integral of a function is a fundamental concept in calculus that represents the accumulation of quantities, such as area under a curve. In simpler terms, if you have a function that describes, for example, the speed of a car over time, the definite integral of that function between two time points would give you the total distance traveled by the car in that time interval.

A definite integral is denoted by \(\int_{a}^{b} f(x) dx\) where \(f(x)\) is the function being integrated, and \(a\) and \(b\) are the limits of integration, representing the interval over which we're summing up the function's values. The result of this operation is a number that provides information such as area, volume, displacement, among others, depending on the context of the function used.

Continuous Functions

A continuous function is one where small changes to the input result in small changes to the output. It has no gaps, jumps, or breaks in its graph. Mathematically, we say that a function \(f\) is continuous over an interval \[a, b\] if it is defined for every point in that interval and does not have any abrupt changes in value.

One of the significant traits of a continuous function is that you can draw its graph without lifting your pencil from the paper. This smoothness is not just visually appealing; it has mathematical implications as well. It allows us to apply the Intermediate Value Theorem and ensures that Riemann sums can converge to the definite integral as the partition gets finer.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) assures us that for every continuous function \(f\), if the function takes two values—let's say \(f(a)\) and \(f(b)\)—it must also take every value between \(f(a)\) and \(f(b)\) at some point within the interval \[a, b\]. This property is naturally intuitive. Imagine drawing a curve without lifting your pencil from start to finish; you will inevitably cross all the heights in between.

The IVT has practical implications for Riemann sums and definite integrals. It ensures that we can find points in each subdivision of a partition where the function's value times the width of the subdivision approximates the area under the curve (or the integral) for that section. As the partitions become very fine, the Riemann sum gets closer and closer to the value of the definite integral.

Properties of Definite Integrals

Definite integrals follow certain properties that make them easier to understand and work with. One essential property is the additivity over intervals. This means if you split an interval \[a, b\] into smaller subintervals, the integral over the entire interval is the sum of the integrals over the subintervals. For instance, \(\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx\) for any \(c\) in \[a, b\].

Also, if a function \(f\) is continuous on \[a, b\], then the integral of \(f\) over \[a, b\] can be determined by examining any collection of subintervals that partition the interval \[a, b\]. This property is instrumental when showing that a Riemann sum for a continuous function over a partition can be equal to the definite integral of that function over the same interval, providing a strong connection between the discrete approximation (Riemann sum) and the continuous truth (definite integral).