Problem 76 - Calculus Early Transcendentals

Question

Suppose \(f\) is continuous on the intervals \([a, p]\) and \((p, b],\) where \(a < p < b\) with a finite jump at \(p .\) Form a uniform partition on the interval \([a, p]\) with \(n\) grid points and another uniform partition on the interval \([p, b]\) with \(m\) grid points, where \(p\) is a grid point of both partitions. Write a Riemann sum for \(\int_{a}^{b} f(x) d x\) and separate it into two pieces for \([a, p]\) and \([p, b] .\) Explain why \(\int_{a}^{b} f(x) d x=\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x\)

Step-by-Step Solution

Answer

Question: Prove that for a function \(f\) which is continuous on the intervals \([a, p]\) and \((p, b]\) with a finite jump at the point \(p\), the integral from \(a\) to \(b\) is equal to the sum of the integrals from \(a\) to \(p\) and from \(p\) to \(b\). In other words, show that \(\int_{a}^{b} f(x) d x=\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x\).

Step 1: Partition Intervals

We will create two uniform partitions. First, partition the interval \([a, p]\) into \(n\) grid points. The width of each sub-interval will be \(\Delta x_1 = \frac{p - a}{n}\). Next, partition the interval \([p, b]\) into \(m\) grid points. The width of each sub-interval will be \(\Delta x_2 = \frac{b - p}{m}\).

Step 2: Riemann Sum for Entire Interval

Since the function \(f\) is not continuous on the entire interval, we cannot directly compute the Riemann sum. However, we can still break down the Riemann sum into two parts that correspond to the continuous intervals. Note that the Riemann sum for the whole interval \([a, b]\) is approximated by the sum of the areas of the rectangles under the graph of the function, which can be written as: $$S = \sum_{i=1}^n f(x_i^*) \Delta x_1 + \sum_{i=1}^m f(y_i^*) \Delta x_2$$ Here, \(x_i^*\) and \(y_i^*\) represent the representative points in the two partitions.

Step 3: Riemann Sum for Each Interval

Now, we can find Riemann sums for each continuous interval separately. For the interval \([a, p]\), the Riemann sum is: $$ S_1 = \sum_{i=1}^n f(x_i^*) \Delta x_1 $$ For the interval \([p, b]\), the Riemann sum is: $$ S_2 = \sum_{i=1}^m f(y_i^*) \Delta x_2 $$

Step 4: Prove the Equality of Integrals

Now, let's show that \(\int_{a}^{b} f(x) d x=\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x\). As we increase the number of grid points in both partitions, \(n\) and \(m\) go to infinity, and the Riemann sums \(S_1\) and \(S_2\) will converge to the true values of their corresponding integrals since \(f\) is continuous on these intervals. Thus, $$ \lim_{n, m \to \infty} S = \lim_{n, m \to \infty} (S_1 + S_2) $$ Since the left-hand side converges to \(\int_{a}^{b} f(x) d x\) and the right-hand side converges to \(\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x\), we can conclude that: $$ \int_{a}^{b} f(x) d x=\int_{a}^{p} f(x) d x+\int_{p}^{b} f(x) d x $$

Key Concepts

Continuous Function

A continuous function is a mathematical function defined by smooth, unbroken curves or lines without any gaps or sudden jumps. When a function is continuous over an interval, it means that for every small change in the input value, the output value also changes smoothly. For integration purposes, this characteristic is crucial as it assures that the function can be graphed on the interval without interruptions.
Continuous functions play a vital role in calculus, especially when computing integrals using techniques like the Riemann Sum. For example, in our exercise, the function \(f\) is continuous on the intervals \([a, p]\) and \((p, b]\). But even if there's a finite jump at \(p\), continuity in the rest of these intervals allows us to break down the area calculation into manageable pieces.
This property of continuity helps us separate the integration process into sub-intervals where \(f\) is smooth and continuous. This makes evaluating each bit easier and allows us to eventually sum them back together to get the integral over the entire range.

Partition of Intervals

Partitioning the interval is an essential step in setting up a Riemann Sum. It involves dividing the overall interval of integration into smaller sub-intervals, making the process of handling large or complex functions manageable.
To create a partition, calculate the width of each sub-interval. For our given intervals \([a, p]\) and \([p, b]\), a uniform partition means each part of these sub-intervals is of equal length. This is determined by dividing each interval into an equal number of grid points, namely \(n\) and \(m\) points, respectively.
For the interval \([a, p]\), the width of each sub-interval is \(\Delta x_1 = \frac{p - a}{n}\). Similarly, for the interval \([p, b]\), the width is \(\Delta x_2 = \frac{b - p}{m}\). These subdivisions essentially help in approximating the total area under the curve of \(f\) by summing up areas of simple geometric shapes, like rectangles, lying within each sub-interval.

Definite Integrals

A definite integral is a mathematical concept that provides the total accumulation of quantities, which, in a geometric sense, is equivalent to the area under a curve between two given points. In integral calculus, the notation \(\int_{a}^{b} f(x) \, dx\) denotes the definite integral of a function \(f\) with respect to \(x\) from \(a\) to \(b\).
Definite integrals allow us to compute the net area between the curve and the x-axis over a specific interval. The exercise's solution showcases the additive property of definite integrals. It demonstrates that you can "split" the problem using intervals: \[ \int_{a}^{b} f(x) \, dx = \int_{a}^{p} f(x) \, dx + \int_{p}^{b} f(x) \, dx \] provided that \(f\) is continuous on the smaller sub-intervals.
The logic behind this lies in how the Riemann Sum approaches the value of the integral. By considering each sub-interval's contribution to the total integral via their own Riemann Sums, you sum the separate parts back to one total sum as the partition becomes finer and more precise. This ultimately validates the integrity of the process in calculating areas under curves accurately, even when faced with a function that's partially continuous.