Problem 87 - Calculus Single Variable

Question

Suppose that \(f\) is continuous on \([a, b]\). Use a substitution to show that $$ \int_{a}^{b} f(x) d x=\int_{0}^{b-a} f(b-x) d x $$

Step-by-Step Solution

Answer

Use substitution \(u = b-x\) to transform the limits and integrand.

Step 1: Understand the Problem

We need to show that the integral of a continuous function over the interval \([a,b]\) can be transformed using a substitution to the integral from \([0, b-a]\) of the same function evaluated at \(b-x\).

Step 2: Define the Substitution

Let's define a substitution which allows us to express this transformation. Let \(u = b-x\), so \(du = -dx\). When \(x = a\), \(u = b-a\), and when \(x = b\), \(u = 0\).

Step 3: Change the Integral Limits

Using the substitution \(u = b-x\), we change the limits of the integral. The integral bounds will change from \([a, b]\) for \(x\) to \([b-a, 0]\) for \(u\).

Step 4: Substitute in the Integral

Insert the substitution into the integral. The integral becomes:\[ \int_{a}^{b} f(x) \, dx = \int_{b-a}^{0} f(u) \, (-du) \].The negative sign from \(du = -dx\) changes the order of the limits.

Step 5: Simplify the Integral

Change the limits from \([b-a, 0]\) to \([0, b-a]\) and incorporate the negative sign:\[ \int_{b-a}^{0} f(u) \, (-du) = \int_{0}^{b-a} f(u) \, du \].

Step 6: Conclude with the Result

Substitute back \(u = b-x\):\[ \int_{0}^{b-a} f(u) \, du = \int_{0}^{b-a} f(b-x) \, dx \].Therefore, the original integral is equal to this transformed integral:

Key Concepts

Continuous Function

A function is said to be continuous if, intuitively, you can draw its graph without lifting your pen from the paper. More formally, a function \( f(x) \) is continuous on an interval \([a,b]\) if for every point \( x = c \) in this interval, the following condition holds:

• For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - c|< \delta \), then \( |f(x) - f(c)| < \epsilon \).

This means that small changes in \( x \) result in small changes in \( f(x) \). As a result, a continuous function has no holes, jumps, or vertical asymptotes within the interval considered.
This property is crucial when dealing with integrals, as we often need to ensure that the function we are integrating is well-behaved across the entire range of integration.

Definite Integral

A definite integral, written as \( \int_{a}^{b} f(x) \, dx \), measures the net area between the function \( f(x) \) and the \( x \)-axis from \( x = a \) to \( x = b \).

• This integral gives an actual numerical value, unlike an indefinite integral which includes a constant of integration \( C \).
• The definite integral encompasses both positive and negative areas, as it accounts for sections above and below the x-axis.

For continuously defined functions over an interval, the definite integral is a powerful tool for measuring total growth, accumulation, or net change through that interval. The solution to these integrals involves taking the antiderivative, or the reverse process of differentiation, of the function over its entire domain between the limits \( a \) and \( b \).

Change of Variables

The change of variables (or substitution) in integration is a technique used to simplify the integration process.
In essence, it's about transforming the possibly complicated original function into a simpler form that is easier to integrate.

• We set a substitution variable, say \( u \), related to the original variable such as \( u = b-x \).
• The differential \( du \) is then derived from this substitution and used to transform the entire integral.

This technique turns a complex integral into a format that's more straightforward. It also often involves changing the integration limits to match the new variable, as shown in the text step-by-step.

Limits of Integration

When calculating a definite integral, the limits of integration define the range over which you're calculating the area under the curve. Also known as bounds, these limits can sometimes appear as constants (like \( a \) and \( b \)) or be functions themselves in more advanced applications.

• After carrying out a substitution in an integral, the original limits of integration need to be recalculated. For example, if \( x = a \) becomes \( u = b-a \) and \( x = b \) becomes \( u = 0 \) after substitution, your new limits change accordingly.
• The process can involve flipping the limits or adjusting them according to the given substitution function \( u \).

In our specific case, after the substitution, the limits of the integral are effectively adjusted to keep the integral equivalent.