Question

Prove: If \(f\) is integrable on \([a, b]\) and \(a=a_{0}

Short answer

Question: Prove that the integral of a function \(f\) over the interval \([a, b]\) can be expressed as the sum of integrals over smaller, non-overlapping subintervals \([a_i, a_{i+1}]\) for i=0, 1, ..., n-1 using the properties of limits and the definition of an integral. Answer: We have proven that the integral of the function \(f\) over the interval \([a, b]\) can be expressed as the sum of integrals over smaller, non-overlapping subintervals \([a_i, a_{i+1}]\) for i=0, 1, ..., n-1 by rewriting the integrals as limits of Riemann sums over refined partitions \(Q_i\), and showing that as \(|| Q_i ||\to 0\), we obtain the desired result.

Step-by-step solution

Define the notation and terms

Let's first make sure we understand the notation and terms used in the problem. We are given a function \(f\) and it is integrable over the closed interval \([a, b]\). We have \(n+1\) points \(a_0, a_1, \cdots, a_n\) such that \(a=a_0<a_1<\cdots<a_n=b\). Our goal is to prove that for such a function and points, the following equation holds: $$ \int_{a}^{b} f(x) d x=\int_{a_{0}}^{a_{1}} f(x) d x+\int_{a_{1}}^{a_{2}} f(x) d x+\cdots+\int_{a_{n-1}}^{a_{n}} f(x) d x . $$

Rewrite as a limit of Riemann sums

The integral of \(f\) can be defined as the limit of Riemann sums over a partition of the interval \([a, b]\). Let \(P\) be a partition for the interval \([a, b]\) that includes points in \(a_0, a_1, \cdots, a_n\), and let \(P_i\) be the partition for the interval \([a_i, a_{i+1}]\). Therefore, we can rewrite the problem using Riemann sums: $$\lim_{||P||\to 0} \sum_{k=0}^{n} \int_{a_k}^{a_{k+1}} f(x) d x = \lim_{|| P_0 ||\to 0} \int_{a_{0}}^{a_{1}} f(x) d x+\cdots+\lim_{|| P_{n-1} ||\to 0} \int_{a_{n-1}}^{a_{n}} f(x) d x.$$

Rearrange the sums

Next, we will rearrange the sums: $$\lim_{||P||\to 0} \left(\sum_{k=0}^{n} \int_{a_k}^{a_{k+1}} f(x) d x\right) = \lim_{|| P_0 ||\to 0} \int_{a_{0}}^{a_{1}} f(x) d x+\cdots+\lim_{|| P_{n-1} ||\to 0} \int_{a_{n-1}}^{a_{n}} f(x) d x.$$

Combine partitions

Now, for each partition \(P_i\), we can find a common partition \(Q_i\) such that \(Q_i\) is a refinement of both \(P\) and \(P_i\). This means that \(P\), \(P_i\), and \(Q_i\) all include the points \(a_k\) for \(k = 0, 1, \cdots, n\). So, we can combine the partitions and write the integral using the refined partitions \(Q_i\): $$\lim_{|| Q_i ||\to 0} \int_{a}^{b} f(x) d x = \lim_{|| Q_0 ||\to 0} \int_{a_{0}}^{a_{1}} f(x) d x+\cdots+\lim_{|| Q_{n-1} ||\to 0} \int_{a_{n-1}}^{a_{n}} f(x) d x.$$

Apply refined partitions

As \(|| Q_i ||\to 0\), we can apply the refined partitions over the given equation. Since all our \(Q_i\) are refinements of both \(P\) and \(P_i\), they have all the points \(a_k\) for \(k = 0, 1, \cdots, n\): $$\int_{a}^{b} f(x) d x = \int_{a_{0}}^{a_{1}} f(x) d x+\cdots+\int_{a_{n-1}}^{a_{n}} f(x) d x.$$ With these steps, we have proven that the integral of the function can be expressed as the summation of integrals over smaller, non-overlapping subintervals.

Key concepts

Riemann Sums

In understanding the integrability of functions, one of the fundamental concepts is Riemann sums. A Riemann sum is essentially an approximation of an integral over an interval, constructed from the sum of function values at certain points multiplied by the distances between these points.

The idea is to divide the interval into several small subintervals, calculate the function's value for each subinterval, and then sum up all these products. The finer the division of the interval (i.e., more subintervals with smaller width), the better the Riemann sum approximates the actual integral. When the maximum width of these subintervals approaches zero, the Riemann sum converges to the integral of the function over the interval, assuming the function is integrable.

Visualizing Riemann Sums

Imagine the curve of a function on a [xmin, xmax] coordinate plane. By constructing a series of rectangles under the curve with heights corresponding to the function's value and bases equivalent to the subintervals' widths, you create a visualization of a Riemann sum.

• If the function's value at the start of the subinterval is chosen, it's called a left Riemann sum.
• Choosing the end of each subinterval results in a right Riemann sum.
• If the maximum function value is selected, you would have an upper Riemann sum.
• Conversely, the minimum value would result in a lower Riemann sum.
• Midpoints can also be considered, called a midpoint Riemann sum.

The difference between the Riemann sum and the integral decreases as the number of subintervals increases, which reflects the concept of the limit and its significance in determining the area under the curve.

Partition of an Interval

A partition of an interval is a sequence of numbers that divides the interval into smaller, non-overlapping subintervals. The partition is central to defining Riemann sums and, consequently, the concept of integration.

A partition is denoted by a set of points, for example, \(a = a_0 < a_1 < \cdots < a_n = b\), within the interval \[a, b\]. The 'fineness' of a partition is determined by the length of the largest subinterval. As the fineness increases, meaning the subintervals become smaller, the partition is better able to capture the nuances of the function's behavior over the interval.

Importance of Partition in Integration

• Partitions allow the creation of Riemann sums, which approximate the integral.
• They help in visualizing and understanding the concept of area under a curve.
• Partitions aid in proving properties of integrals, like additivity over intervals.

In the context of the example problem, the partition outlined by the points \(a_0, a_1, \ldots, a_n\) is crucial because it sets the stage for breaking the entire integral into smaller, more manageable integrals over each subinterval.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value. The concept of a limit is vital for understanding continuity, derivatives, and integrals of functions. It's the cornerstone upon which the entire architecture of calculus is built.

In the context of Riemann sums and integrals, the limit is used to describe the process of making the sum of the areas of the rectangles arbitrarily close to the actual area under the curve. Mathematically, if we take a Riemann sum \( S_P(f) \) associated with a partition \( P \) of \[a, b\], the integral \( \int_a^b f(x)dx \) is defined as the limit of these sums when the width of the largest subinterval in \( P \) approaches zero.

Significance of Limits in Integrals

• Limits enable the transition from discrete sums to continuous integrals.
• They provide the precise definition of the definite integral as the limit of Riemann sums.
• The existence of a limit of a function over an interval is a precondition for its integrability on that interval.

Using the limit in the example provided, it is shown that as the mesh of partitions gets finer, the Riemann sums converge to the definite integral, which is compartmentalized into the sum of integrals over smaller intervals.