Question

Show that if \(f(x)\) is continuous for \(0 \leq x \leq 1\), then $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x . $$

Short answer

To summarize, we have shown that the given limit of the sum is equivalent to the Riemann sum over the interval [0,1] for the continuous function f(x). Therefore, by the definition of Riemann sums, this limit converges to the definite integral of f(x) over the same interval, thus proving the equality: $$ \lim_{n\rightarrow \infty}\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) dx. $$

Step-by-step solution

Recall the definition of Riemann sums

A Riemann sum is a way to approximate the definite integral using finite sums. Let a continuous function \(f(x)\) be defined on the interval \([a, b]\), and partition this interval into \(n\) equal subintervals, each of length \(\Delta x = \frac{b-a}{n}\). Then the Riemann sum is given by: $$ \sum _{i=1}^{n} f(x_i^*) \Delta x, $$ where \(x_i^*\) is a point in the \(i\)-th subinterval. As \(n \rightarrow \infty\), this sum converges to the definite integral of \(f(x)\) from \(a\) to \(b\), that is: $$ \lim_{n\rightarrow \infty} \sum _{i=1}^{n} f(x_i^*) \Delta x = \int_{a}^{b} f(x) dx. $$

Connect the given limit with Riemann sums

We want to show that $$ \lim_{n\rightarrow \infty}\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) dx. $$ Observe that the sum in the limit expression is a Riemann sum. The interval \([0,1]\) is partitioned into \(n\) equal subintervals, each with length \(\Delta x=\frac{1}{n}\). Also, each term \(f\left(\frac{i}{n}\right)\) corresponds to the evaluation of the function at a point in the \(i\)-th subinterval. We just need to write the sum in the limit expression as a Riemann sum.

Rewrite the limit expression as a Riemann sum

We have the following limit expression: $$ \lim_{n\rightarrow \infty}\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f\left(\frac{n}{n}\right)\right]. $$ Now, we rewrite this expression as a Riemann sum. Notice that the length of each subinterval is \(\Delta x =\frac{1}{n}\), and the function is evaluated at \(x_i^*=\frac{i}{n}\), where \(i=1,2,\dots,n\). Therefore, we can rewrite the given expression as the following Riemann sum: $$ \lim_{n\rightarrow \infty}\sum_{i=1}^{n}f\left(\frac{i}{n}\right) \frac{1}{n}. $$

Conclude the proof

Since we have shown that the given limit is equivalent to a Riemann sum, we can now apply the Riemann sum definition and obtain the definite integral of the function over the interval \([0,1]\): $$ \lim_{n\rightarrow \infty}\sum_{i=1}^{n}f\left(\frac{i}{n}\right) \frac{1}{n}=\int_{0}^{1}f(x)dx. $$ This completes the proof.

Key concepts

Definite Integral

Understanding the definite integral is essential for solving problems in calculus. The definite integral of a function, typically denoted as \( \int_{a}^{b} f(x) \, dx \), represents the total accumulated value of the function \( f(x) \) over the interval \([a, b]\). This topic is crucial in determining areas under curves and has applications across physics and engineering.

• The definite integral ties closely with Riemann sums, which approximate the area by summing up areas of small rectangles under the curve.
• As the partition becomes finer (i.e., as the number of rectangles increases), the sum converges to the exact area.
• The Fundamental Theorem of Calculus connects the derivative with the integral, making it a powerful tool for evaluating integrals.

By understanding definite integrals, students can better grasp the entire calculus landscape, discovering how accumulation and rates connect through these concepts.

Continuity of Functions

Continuity of functions is a foundational aspect of calculus. A function \(f(x)\) is continuous on an interval if it can be graphed without lifting the pen from the paper. For technical purposes, continuity at a point \(c\) in its domain means that:

• The function is defined at \(c\): \( f(c) \) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists: \( \lim_{x \to c} f(x) \) exists.
• The limit equals the function value: \( \lim_{x \to c} f(x) = f(c) \).

This means any small change in \(x\) results in a small change in \(f(x)\), making it crucial for defining integrals and derivatives.
Continuity on closed intervals, like \([0, 1]\), ensures functions behave predictably, allowing us to apply concepts like the Intermediate Value Theorem and ensuring the existence of limits in Riemann sums.

Limit of a Sequence

The concept of the limit of a sequence is fundamental in calculus, representing the value that the terms of a sequence "approach" as the number of terms goes to infinity. Mathematically, if \( \{a_n\} \) is a sequence, \( \lim_{n \to \infty} a_n \) is the limit if for any small number \( \epsilon > 0\), there exists an \(N\) such that for all \(n > N\), \(|a_n - L| < \epsilon\), where \(L\) is the limit. This often means the sequence steadily gets closer to \(L\) as \(n\) becomes very large.
Applying limits is crucial in evaluating Riemann sums, as shown in the original problem, where taking the limit of the sums gives the exact value of the definite integral.
Understanding limits allows students to analyze how functions behave as they extend towards infinity, laying the foundation for the rigorous study of calculus.

Partition of an Interval

A partition of an interval is a way to break it down into smaller subintervals, which is essential for understanding Riemann sums and integrals. In the interval \([a, b]\), a partition divides it into subintervals \([x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]\), where \(a = x_0 < x_1 < \cdots < x_n = b\).

• The choice of subdivision points affects the construction of the Riemann sum and ultimately the approximation of the integral.
• Each subinterval can have a length \(\Delta x_i = x_{i+1} - x_i\), and if each subinterval is equal, then \(\Delta x = \frac{b-a}{n}\).
• In calculus, as the mesh or norm of the partition (the size of the largest subinterval) decreases to zero, the Riemann sum converges to the definite integral of the function over \([a, b]\).

Having a solid understanding of partitions helps students grasp the process of integration, enabling them to approximate and compute exact areas by refining partitions to make accurate estimations.