Question

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 where \(f(x)=\sin x, a=0, b=\pi,\) and \(c=1 / 2\)

Short answer

In this problem, we used the substitution method to prove that for a given change of variables, \(u = cx\), the integral equation holds true: $$\int_{a}^{b} f(cx) dx = \frac{1}{c}\int_{ac}^{bc} f(u) du$$ We also illustrated this change of variables using a specific example with \(f(x) = \sin x, a = 0, b = \pi,\) and \(c = \frac{1}{2}\) by graphing the original function and the new function after performing the change of variables, showing the geometric interpretation of the scaling factor.

Step-by-step solution

Define the substitution

Let \(u = cx\), where \(c\) is a real number. We will differentiate \(u\) with respect to \(x\) to find \(\frac{du}{dx}\): $$\frac{du}{dx} = \frac{d(cx)}{dx} = c$$ Now, we find \(dx\) in terms of \(du\): $$dx = \frac{1}{c}du$$

Substitute and change the limits of integration

We will substitute \(u\) into the original integral and change the limits of integration: $$\int_{a}^{b} f(cx) dx = \int_{ac}^{bc} f(u) \cdot \frac{1}{c} du$$

Simplify the integral

Now simplify the integral by taking the constant \(\frac{1}{c}\) outside the integral: $$\frac{1}{c}\int_{ac}^{bc} f(u) du$$

Proving the equation

We have now shown that the integral after substitution is indeed equal to the given equation: $$\int_{a}^{b} f(cx) dx = \frac{1}{c}\int_{ac}^{bc} f(u) du$$

Illustrate the change of variables

Now we will draw a picture to illustrate the change of variables for the specific case \(f(x)=\sin x, a=0, b=\pi,\) and \(c=\frac{1}{2}\). 1. Graph the function \(y = \sin x\) on the interval \([0, \pi]\). 2. Perform the change of variables: \(u = \frac{1}{2}x\). 3. Graph the function \(y = \sin \frac{1}{2}x\) on the interval \([0, 2\pi]\). 4. Notice that the new function \(y = \sin \frac{1}{2}x\) stretches horizontally by a factor of 2. This visualization helps us understand how the given change of variables affects the integral, and the geometric interpretation of the scaling factor.

Key concepts

Scaling Transformation

A scaling transformation involves altering the size of a geometric figure while maintaining its shape. In the realm of integrals, this transformation affects the input to a function, leading to a change in the integral's range and evaluation. When we apply a transformation like

• Scaling: This means adjusting the function by a constant factor, which modifies how the entire function behaves.
• For instance, using the substitution \( u = cx \), where \( c \) is a constant, stretches or shrinks the graph of \( f(x) \) horizontally, depending on whether \( c \) is greater than or less than 1.

In essence, when you scale a function, you're altering the space on which it operates. Consider the specific transformation applied to the function \( f(x) = \sin x \), from interval \([0, \pi]\). If \( c = \frac{1}{2} \), the new function becomes \( \sin(\frac{1}{2}x) \) on the interval \([0, 2\pi]\). This change effectively stretches the sine wave, illustrating how scaling affects the integral.

Integral Substitution

Integral substitution, often dubbed as the variable substitution, is a pivotal technique in calculus that helps simplify the computation of integrals. By replacing a variable in the function with another expression, this method can convert a complicated integral into a simpler one.
Here's how it works:

• By letting \( u = cx \), you introduce a new variable, \( u \), to simplify the integral calculation.
• Consequently, \( du = c\, dx \), which means \( dx = \frac{1}{c}\, du \).
• The substitution simplifies the integral, allowing you to focus on smaller, more manageable parts.

Using substitution can make a formidable integral far more approachable. For example, applying the substitution \( u = \frac{1}{2}x \) simplifies evaluating \( \int_0^{\pi} \sin(\frac{1}{2}x) \, dx \) by transforming it into \( \frac{1}{2}\int_0^{2\pi} \sin(u) \, du \). This change makes integration much more convenient.

Geometric Interpretation

The geometric interpretation of changes in variables provides a visual understanding of how transformations modify functions and their integrals. Consider the given example of scaling with the function \( f(x) = \sin x \). Visualizing this transformation can offer valuable insight into the effect of variable changes.
Here's what happens:

• Drawing the curve \( y = \sin x \) on the interval \([0, \pi]\) shows a single full wave.
• When scaling is applied using \( u = \frac{1}{2}x \), the interval of integration becomes \([0, 2\pi]\), effectively doubling the length over which the function is plotted and therefore stretching the sine wave horizontally.
• Thus, the resultant function \( y = \sin(\frac{1}{2}x) \) now features a broader wave pattern spanning \([0, 2\pi]\).

This visualization aids in comprehending how integrals adjust boundaries and shapes in response to variable changes. It effectively uses a "stretch" of the space and demonstrates how integrals' calculations reflect these transformations, providing a concrete geometric perception.