Question

Prove: If \(\int_{a}^{b} f(x) d x\) exists as a proper or improper integral, then $$ \lim _{x \rightarrow b-} \int_{x}^{b} f(t) d t=0 $$

Short answer

Question: If the integral of f(x) exists as a proper or improper integral on the interval [a, b], prove that the limit as x approaches b from the left of the integral from x to b of f(t) dt is equal to 0.

Step-by-step solution

Assumption

The integral \(\int_{a}^{b} f(x) d x\) exists as either a proper or improper integral. This means that there is a finite value for the integral. Let this value be F, i.e. $$ F= \int_{a}^{b} f(x) dx $$

Consider a subinterval [x, b]

Let's consider a subinterval [x, b] and the integral of f(t) on this interval, as x approaches b from the left. We have: $$ \int_{x}^{b} f(t) dt $$ Note that when x = a, this integral represents the same as the original integral and should equal F.

Evaluate the difference of two integrals

To evaluate the limit as x approaches b from the left, we can evaluate the difference between the original integral and the integral from step 2: $$ \lim_{x \rightarrow b-} \left( F - \int_{x}^{b} f(t) dt \right) $$ This difference represents the area under the curve of f(t) from a to x.

Explain using the concept of area

As x approaches b, the integral from x to b represents the area under the curve that is getting smaller and smaller, since the interval is shrinking. The area is approaching zero because the width of the interval is approaching zero as x gets closer to b.

Evaluate the limit

Taking the limit as x approaches b from the left, we can rewrite the difference expression as: $$ \lim_{x \rightarrow b-} \left( F - \int_{x}^{b} f(t) dt \right) = \lim_{x \rightarrow b-} \left( 0\right) $$ Because the area is approaching zero as x gets closer to b, the limit evaluates to 0: $$ \lim _{x \rightarrow b-} \int_{x}^{b} f(t) d t=0 $$ Hence, we have proved that if the integral of f(x) exists as a proper or improper integral on the interval [a, b], the limit as x approaches b from the left of the integral from x to b of f(t) dt is equal to 0.

Key concepts

Limit of an Integral

When we talk about the limit of an integral in the context of real analysis, we're entering a realm where calculus and mathematical precision intersect. The idea is to understand what happens to the value of an integral as we adjust the limits of integration. In particular, this concept involves investigating the behavior of integrals as the upper limit approaches a certain value.

To make it concrete, consider the integral \( \int_{a}^{b} f(x) dx \). We are interested in seeing what happens as we take the upper limit and inch it ever closer to the value of 'b' from the left. This is expressed mathematically as the limit \( \lim_{x \rightarrow b-} \int_{x}^{b} f(t) dt \). Intuitively, as the upper limit approaches 'b', the interval on which we are integrating becomes smaller, and so the integral — which represents the area under the curve of \( f(x) \) — shrinks accordingly.

Link to Convergence

For students, a useful way to think about this is to visualize a graph where the 'x' is moving toward 'b'. As 'x' gets closer, the area under the curve between 'x' and 'b' gets smaller until it becomes negligible. If the original integral \( \int_{a}^{b} f(x) dx \) is convergent, meaning it has a finite value, then as 'x' gets infinitesimally close to 'b', the integral of the shrunk interval (from 'x' to 'b') must also converge to a specific value. And in our case, that specific value is zero, which corresponds to the fact that there's virtually no area left under the curve in that interval.

Real Analysis

The field of real analysis is a branch of mathematics that deals with the behavior of real numbers, sequences and series of real numbers, and real-valued functions. One of the key aspects of real analysis is the study of integrals and limits — precisely what we're looking at when we consider the limit of an integral.

In real analysis, we rigorously define what it means for a sequence or function to converge, and apply these definitions to understand the behavior of integrals like \( \int_{a}^{b} f(x) dx \) over different intervals. The 'proofs' aspect of real analysis is significant here—demonstrating formally that the limit of the integral shrinks to zero as the upper limit approaches 'b' is part and parcel of the subject's analytical nature.

Understanding Convergence

One of the central concepts in real analysis is convergence. In simple terms, a function or sequence converges if it approaches a specific value as it extends to infinity or, in the case of functions, as the input approaches a certain point. The proof you’re looking at is a classic example of demonstrating convergence: it shows that the integral over a diminishing interval converges to zero, which is coherent with the fundamental properties of definite integrals in real analysis that dictate the outcome of such limits.

Area Under a Curve

When people first learn about integrals in calculus, one of the most visually intuitive concepts is that of the area under a curve. A definite integral like \( \int_{a}^{b} f(x) dx \) essentially measures the total area under the curve of the function \( f(x) \) from \( x=a \) to \( x=b \) along the x-axis.

This geometric interpretation helps to visualize many calculus phenomena, including the problem at hand. The crux of the exercise is to realize that as 'x' marches closer to 'b', the interval over which we're measuring this area shrinks. Imagine a rectangle under the curve where one side gets shorter and shorter. Eventually, this rectangle becomes a line as its width goes to zero, and thus the area — which is the product of length and width — also goes to zero.

Visualizing the Problem

For an improved understanding, consider graphing the function \( f(x) \) and shading the area from \( x \) to 'b' as \( x \) moves closer to 'b'. Watch the shaded region get smaller and ultimately vanish, aligning with the principle that the limit of the integral over a narrowing interval trends towards zero. This ties back to the fundamental theorem of calculus, which relates the concept of the derivative to the area under a curve. It’s a principle that has wide-ranging implications across mathematics and physics, and provides the foundation for understanding integration and the behavior of areas as intervals change.