Question

Suppose that \(\int_{a}^{b} f(x) d x\) exists and there is a number \(A\) such that, for every \(\epsilon>0\) and \(\delta>0,\) there is a partition \(P\) of \([a, b]\) with \(\|P\|<\delta\) and a Riemann sum \(\sigma\) of \(f\) over \(P\) that satisfies the inequality \(|\sigma-A|<\epsilon .\) Show that \(\int_{a}^{b} f(x) d x=A\).

Short answer

Question: Prove that if a given Riemann sum of a function \(f(x)\) on an interval \([a, b]\) approaches a specific number \(A\) for every \(\epsilon > 0\) and \(\delta > 0\), then the value of \(\int_{a}^{b} f(x)dx\) equals that number \(A\). Answer: To prove this, we showed that the given condition implies that the function \(f(x)\) meets the definition of a Riemann integrable function. Since the function is Riemann integrable, the value of the integral \(\int_{a}^{b} f(x)dx\) equals the number \(A\), which satisfies the specified conditions for every \(\epsilon > 0\) and \(\delta > 0\).

Step-by-step solution

Definition of a Riemann integrable function

A function \(f(x)\) is said to be Riemann integrable on an interval \([a, b]\) if there exists a number \(A\) such that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for any partition \(P\) of \([a, b]\) with \(\|P\| < \delta\) and for any Riemann sum \(\sigma\) of \(f\) over the partition \(P\), the inequality \(|\sigma - A| < \epsilon\) holds.

Reiterate given condition

We are given that for every \(\epsilon > 0\) and \(\delta > 0\), there exists a partition \(P\) of \([a, b]\) with \(\|P\| < \delta\) and a Riemann sum \(\sigma\) of \(f\) over the partition \(P\) such that the inequality \(|\sigma - A| < \epsilon\) is satisfied.

Show that \(f\) is Riemann integrable

To demonstrate that \(f\) is Riemann integrable, we must show that the given condition implies that the function meets the definition of a Riemann integrable function. Let \(\epsilon > 0\) be given. Since the given condition holds for every \(\delta > 0\), we can choose a \(\delta > 0\). Then, by the given condition, there exists a partition \(P\) of \([a, b]\) with \(\|P\| < \delta\) and a Riemann sum \(\sigma\) of \(f\) over the partition \(P\) such that the inequality \(|\sigma - A| < \epsilon\) is satisfied. This is precisely the definition of a Riemann integrable function, and we have satisfied it with the given condition.

Conclusion

Since \(f\) has been shown to satisfy the definition of a Riemann integrable function, we can conclude that \(\int_{a}^{b} f(x)dx = A\), where \(A\) is the number that satisfies the specified conditions for every \(\epsilon > 0\) and \(\delta > 0\).

Key concepts

Riemann sum

A Riemann sum is a method for approximating the total area under a curve, which represents the integral of a function over an interval. The idea behind a Riemann sum is to divide the region under a curve into small shapes—usually rectangles—and then sum up the areas of these shapes to get an approximation of the total area. This is particularly important for functions that are not easily integrable using basic calculus techniques.

In practice, a Riemann sum can be calculated by choosing specific points in each subinterval of a partition and evaluating the function at these points. The function values are then multiplied by the width of the subintervals, and all products are summed up to provide the approximation:

\[\sigma = \sum_{i=1}^{n} f(x_i^*) \Delta x_i\]

Where \(x_i^*\) is a point in the subinterval, and \(\Delta x_i\) is the width of the subinterval. Different types of Riemann sums exist, such as left, right, and midpoint sums, depending on the point \(x_i^*\) chosen.

The accuracy of a Riemann sum approximation can be controlled by the partition, with smaller partitions leading to more accurate approximations of the integral.

partition of an interval

Partitioning an interval is an essential step in calculating a Riemann sum. A partition of an interval \([a, b]\) is a set of points that divide the interval into smaller subintervals. Mathematically, a partition \(P\) can be represented as:

\[P = \{a = x_0, x_1, x_2, ..., x_n = b\}\]

where each \(x_i\) is a point in the interval, and \(n\) represents the number of subintervals.

A finer partition means having more subintervals, which results in a smaller maximum width of the subintervals, denoted by \(\|P\|\). In the context of Riemann integrability, the partition is crucial because a finer partition (which means \(\|P\|\) is smaller) allows for better approximation of the function's integral. The selection of partitions is tied directly to achieving a desired degree of accuracy in the Riemann sum approximation.

Choosing the right partition is a delicate balance between computational efficiency and the desired precision of the integral approximation.

epsilon-delta condition

The epsilon-delta condition is a cornerstone of the formal definition of limits and integrability in mathematics. It provides the framework for understanding how closely a Riemann sum approximates an integral. Essentially, it describes how small changes in the partition size can affect how accurately a Riemann sum approximates the integral.

For a function to be Riemann integrable on an interval \([a, b]\), we use the epsilon-delta criterion. This criterion states that for every \(\epsilon > 0\)—which is a measure of the desired accuracy—there exists a \(\delta > 0\) such that for any partition \(P\) with \(\|P\| < \delta\), the Riemann sum \(\sigma\) satisfies the following condition:

\[|\sigma - A| < \epsilon\]

where \(A\) is the exact value of the integral of the function over \([a, b]\).

This condition ensures that by choosing a sufficiently fine partition, the Riemann sum can be made arbitrarily close to the actual integral, ensuring the function is Riemann integrable. This precision control is vital in confirming the functionality of the Riemann integral.