Problem 105 - Calculus Early Transcendentals

Question

Substitution: scaling Another change of variables that can be interpreted geometrically is the scaling \(u=c x,\) where \(c\) is a real number. Prove and interpret the fact that $$\int_{a}^{b} f(c x) d x=\frac{1}{c} \int_{a c}^{b c} f(u) d u$$ Draw a picture to illustrate this change of variables in the case that \(f(x)=\sin x, a=0, b=\pi,\) and \(c=\frac{1}{2}\)

Step-by-Step Solution

Answer

#Short Answer# The given integral transformation holds true after performing the substitution \(u = cx\). The transformation allows the integration of a compressed or stretched version of a function over a specific interval, resulting in the same value as integrating the original function over a different interval. In the given example, integrating the sine function \(f(x)=\sin{x}\) compressed by a factor of \(\frac{1}{2}\) between \(0\) and \(\pi\) results in the same value as integrating the original sine function between \(0\) and \(\frac{\pi}{2}\). The transformation \(u=cx\) serves as a method to adjust the function and its integration bounds while preserving the value of the integral.

Step 1: Perform the substitution

In order to prove the given integral transformation, we need to perform a substitution: let \(u = cx\). Differentiate both sides of the equation with respect to \(x\) to get \(\frac{du}{dx} = c\), and from this, \(dx = \frac{1}{c}du\). Evaluate the new limits of integration based on the substitution: \({a c}^{b c}\).

Step 2: Substitute variables and limits into the integral

Now substitute these new variables and limits into the integral: $$\int_{a}^{b} f(cx) dx = \int_{ac}^{bc} f(u) \frac{1}{c}du$$

Step 3: Simplify the integral

Since \(c\) is a constant, we can move it outside the integral, which results in the desired equality: $$\int_{a}^{b} f(cx)dx = \frac{1}{c} \int_{ac}^{bc} f(u) du$$ The given integral transformation is proved to be true.

Step 4: Draw a picture illustrating the change of variables

In this specific example, consider the function \(f(x) = \sin x\), with \(a = 0\), \(b = \pi\), and \(c = \frac{1}{2}\). Then, the left side integral becomes: $$\int_{0}^{\pi} \sin\left(\frac{1}{2}x\right) dx$$ The transformation changes the integral to the following: $$\frac{1}{\frac{1}{2}} \int_{0 \cdot \frac{1}{2}}{\pi \cdot \frac{1}{2}} \sin u du = 2 \int_{0}^{\frac{\pi}{2}} \sin u du$$ The function \(\sin(\frac{1}{2}x)\) represents a horizontally compressed version of the sine function, while the transformed integral \(\sin u\) represents the original sine function. Integrating the compressed sine function between \(0\) and \(\pi\) gives the same value as integrating the original sine function between \(0\) and \(\frac{\pi}{2}\), meaning that the transformation \(u=c x\) acts like a compression or stretching of the function along the \(x\)-axis.

Key Concepts

Change of Variables in Integration

The change of variables, often referred to as variable substitution, is a powerful technique in integration that allows us to simplify a given integral into a form that is easier to solve. Think of it like translating a complex text into a simpler language that you're more comfortable with. For example, when integrating a composite function such as f(cx), we can introduce a new variable u to replace cx, then express dx in terms of du. This process essentially rewrites the integral in terms of u, making it easier to compute.

Consider this transformation as a kind of mathematical alchemy where we transform one integral into another, potentially simpler, integral. It's like converting gold into silver—not in terms of value, but in terms of utility; sometimes, silver is simply easier to work with!

In practical terms, after setting u = cx, we find du/dx = c, which lets us rewrite dx as du/c. The integration limits also change accordingly, from a to b in the x-domain to ac to bc in the u-domain. This technique not only simplifies the integral but also can reveal interesting properties about the function being integrated.

Integral Scaling

Integral scaling deals with the alteration of an integral's domain through multiplication by a scalar constant, c. Imagine a rubber band representing the interval from a to b. If you stretch or compress this rubber band by a factor of c, you're visually performing integral scaling.

Mathematically, scaling the argument of a function within an integral by a constant c leads us to multiply the integral by 1/c. This reflects the fact that stretching the interval wider (when c > 1) dilutes the effect of the function over a larger area, while compressing it (when c < 1) concentrates the function's influence. Hence, our scaling factor inversely affects the integral's value.

Why Use Scaling?

• Simplify the integral.
• Reveal symmetry or periodic properties.
• Facilitate comparison between related functions.

Whether we’re expanding or compressing our integral's domain, considering the stretching or shrinking of the functional input via scaling is key to mastering integration techniques.

U-Substitution

U-substitution, in essence, is the integration counterpart to the chain rule in differentiation. It is employed when faced with an integral that is difficult to evaluate directly. Think of u-substitution as a tool for untangling a knotted rope so that you can straighten it out and measure its length more easily.

When applying u-substitution, one identifies a part of the integrand that can be replaced with a new variable u. The differential dx is then expressed in terms of du, making it possible to rewrite the original integral in a form that is more approachable.

Steps for U-Substitution:

• Pick a u that simplifies the given integral when substituted.
• Differentiate u to find du and solve for dx.
• Substitute u and dx into the integral.
• Change the limits of integration if necessary.
• Integrate with respect to u.
• Replace u with the original variables to find the final solution.

By mastering u-substitution, you unlock the ability to streamline integrals and unveil their solutions.

Integration Limits Transformation

Integration limits transformation is like adjusting the zoom on a camera lens to get the perfect shot; it shifts the perspective from which we view the integral's domain. This is particularly important when we've performed a substitution that changes the variable of integration.

For a definitive integral, the limits represent the interval over which we are summing our function's values. When we substitute u = cx, we alter this interval to reflect our new variable. The original limits in terms of x are transformed into limits in terms of u, ensuring that we evaluate the integral over the correct interval.

Let's say our original lower limit was a and upper limit b. After the substitution, these become ac and bc respectively. This means we adjust our analytical lens, viewing the function's behavior from a different scope. Remember, it's crucial to apply this change correctly; failure to transform the limits could lead to evaluating the integral over an incorrect interval, drastically changing the result.

Always consider these new limits as the frame within which the integral's narrative unfolds. By carefully managing the frames' positions, we maintain the integrity of the integral's story.