Question

a. Graph the functions \(f_{1}(x)=\sin ^{2} x\) and \(f_{2}(x)=\sin ^{2} 2 x\) on the interval \([0, \pi] .\) Find the area under these curves on \([0, \pi]\) b. Graph a few more of the functions \(f_{n}(x)=\sin ^{2} n x\) on the interval \([0, \pi],\) where \(n\) is a positive integer. Find the area under these curves on \([0, \pi] .\) Comment on your observations. c. Prove that \(\int_{0}^{\pi} \sin ^{2}(n x) d x\) has the same value for all positive integers \(n\) d. Does the conclusion of part (c) hold if sine is replaced by cosine? e. Repeat parts (a), (b), and (c) with \(\sin ^{2} x\) replaced by \(\sin ^{4} x\) Comment on your observations. f. Challenge problem: Show that, for \(m=1,2,3, \ldots\) $$\int_{0}^{\pi} \sin ^{2 m} x d x=\int_{0}^{\pi} \cos ^{2 m} x d x=\pi \cdot \frac{1 \cdot 3 \cdot 5 \cdots(2 m-1)}{2 \cdot 4 \cdot 6 \cdots 2 m}$$

Short answer

In this exercise, we have explored the properties of functions involving even powers of trigonometric functions, specifically sine and cosine. We have graphed these functions, calculated the areas under their curves using integration, and analyzed how these areas are related to positive integer values of n. We have also proved that the integral has the same value for all positive integers n for both sine and cosine, and extended the analysis to \(\sin^4x\). Finally, we have solved the challenge problem by proving a generalized formula for these types of integrals. The key concepts involved in this exercise include graphing trigonometric functions, calculating areas under curves using integration, and proving general formulas.

Step-by-step solution

Graph \(f_{1}(x) = \sin^{2}x\) and \(f_{2}(x) = \sin^{2}2x\)

To graph these functions, we can recognize their shapes as sinusoidal waves. A sine wave is oscillating between 0 and 1, while its square retains the oscillating behavior but is always positive. The function \(f_{2}(x)\) has a doubled frequency as compared to \(f_{1}(x)\), which means it oscillates twice as fast. Graph the functions on a plotting tool within the given interval \([0, \pi]\).

Find the areas under the curves using integration

The area under the curve can be found using integration. We can find the areas separately for \(f_{1}(x)\) and \(f_{2}(x)\) first: - Area under \(f_{1}(x)\): \(\int_{0}^{\pi}\sin^{2}x dx\) - Area under \(f_{2}(x)\): \(\int_{0}^{\pi}\sin^{2}2x dx\) Once the areas have been calculated, we will have the answer to part (a). #b. Graph a few more functions on \([0, \pi]\) and comment#

Graph more functions \(f_{n}(x) = \sin^{2}nx\)

Choose any positive integer \(n\) and graph the function \(f_{n}(x) = \sin^{2}(nx)\). For example, we can choose \(n = 3\), and \(n = 4\). Observe the shapes of the graphs and make any observations about their patterns, frequency, and amplitudes.

Find the areas under the curves and comment

Calculate the areas under the curves for different \(n\), just like in part (a), by calculating the integral: - \(\int_{0}^{\pi}\sin^{2}(nx) dx\) You can also use the equality found in part a between sine functions with n=1 and n=2 as a guide. Compare the areas for different values of \(n\) and make any observations or generalizations about the relationship between \(n\) and the area. #c. Prove that the integral has the same value for all positive integers \(n\)#

Setup integral identity

To prove the integral is the same, we will start with the formula derived in Step 2 of part (b): $$\int_{0}^{\pi} \sin^{2}(n x) dx$$ Our objective is to prove that this formula has the same value for all positive integer \(n\).

Prove the equation holds for all n

With the integral expression, use the trigonometric identity \(1 - \cos(2x) = 2\sin^2(x)\) and sum-to-product formulas to show that the integral indeed evaluates to the same value for every positive integer \(n\). This will prove that the given integral is the same for all positive integers \(n\). #d. Check if the conclusion holds if sine is replaced by cosine# Repeat the process from part (a), (b), and (c) by replacing \(\sin^2(nx)\) with \(\cos^2(nx)\). Calculate the area by integrating \(\cos^2(nx)\) over \([0, \pi]\). Observe if the area calculated is the same as in part (c). #e. Repeat parts (a), (b), and (c) with \(\sin^{2} x\) replaced by \(\sin^{4} x\)#

Graph the new functions and find area under the curves

Repeat the same steps as explained earlier in parts (a) and (b) but replacing the sine functions with the 4th power of them. Observe the patterns and compare the results for different functions.

Prove the integral formula for the new functions

To prove the integral is the same for the new functions, setup and prove the integral for the expression: $$\int_{0}^{\pi} \sin^{4}(n x) dx$$ Show that this integral evaluates to the same value for every positive integer \(n\). #f. Solve the challenge problem#

Prove the integral formula for \(m = 1,2,3,...\)

In order to prove the formula, we can start with the general expression: $$\int_{0}^{\pi} \sin^{2m}(x) dx = \int_{0}^{\pi} \cos^{2m}(x) dx$$ This can be proven for all \(m\) by applying reduction formulas and using properties of the gamma function \(\Gamma(x)\) with the iterated sines and iterative integrals.

Calculate the integral using the given formula

Lastly, we can calculate the integral using the formula provided in the exercise: $$\pi \cdot \frac{1 \cdot 3 \cdot 5 \cdots(2m-1)}{2 \cdot 4 \cdot 6 \cdots 2m}$$ This calculation will evaluate and confirm the integral results for the challenge problem.

Key concepts

Graphing Trigonometric Functions

Graphing trigonometric functions such as \( f_{1}(x)=\sin^{2}x \) and \( f_{2}(x)=\sin^{2}(2x) \) can initially seem daunting, but it's all about observing patterns. These functions are periodic with oscillatory behavior. The square of the sine function, \( \sin^{2}x \), will only provide values between 0 and 1.When graphing, you will notice that \( f_{2}(x) \) oscillates twice as fast as \( f_{1}(x) \), owing to its increased frequency. This means it completes more cycles within the same interval \( [0, \pi] \).
This increased number of oscillations, or frequency, is a crucial observation when analyzing trigonometric functions. When you graph more functions of the form \( f_{n}(x)=\sin^{2}(nx) \) with positive integer values for \( n \), the patterns remain consistent in terms of frequency effects; that is, higher values of \( n \) equate to more oscillations within a given segment. Understanding these patterns empowers you to predict behavior and sketch trigonometric graphs more intuitively.

Integration Techniques

Integration techniques are essential tools for finding the area under curves, especially with trigonometric functions. When tasked to find the area under \( f_{1}(x)=\sin^{2}(x) \) from \( 0 \) to \( \pi \), we employ integration. Specifically, we compute \( \int_{0}^{\pi} \sin^{2}(x) \, dx \).
To solve this, we use a trigonometric identity to simplify \( \sin^2(x) \) to a more integrable form, \( \frac{1 - \cos(2x)}{2} \). This conversion highlights an important step using trigonometric identities and allows for straightforward integration. Similarly, this technique extends to functions like \( \sin^{2}(2x) \) by setting up the corresponding identity and integrating over the specified domain.
By following these methods, you can tackle any integral involving squared sine or cosine functions with confidence. It's a matter of knowing your identities and then integrating with respect to the variable. Practicing these techniques will greatly improve your problem-solving skills.

Trigonometric Identities

Trigonometric identities are fundamental tools in simplifying complex expressions. Especially during integration, knowing identities like \( 1 - \cos(2x) = 2\sin^2(x) \) proves invaluable. These identities transform integrals into more manageable forms.
For instance, when finding \( \int_{0}^{\pi} \sin^{2}(x) \, dx \), using the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) simplifies the function to a basic polynomial that is easy to integrate. Remember, trigonometric identities are not only used for simplification but are crucial in proving relationships between different trigonometric functions.
As you proceed with exercises, practice recognizing which identities will best simplify the problem at hand. These identities are like keysthat unlock solutions to seemingly difficult problems. The more you practice, the more natural it becomes to apply these identities efficiently.

Reduction Formulas

Reduction formulas are systematic tools that simplify the integration of powers of sine and cosine functions. They're particularly useful when dealing with higher powers of trigonometric functions. For example, consider proving that \( \int_{0}^{\pi} \sin^{2m}(x) \ dx \) is equal to \( \int_{0}^{\pi} \cos^{2m}(x) \ dx \). Using reduction formulas, this is made possible by expressing \( \sin^{2m}(x) \) in terms of reducible components, then successively applying the reduction formulas for each integral step.
The challenge problem involving these integrals and identifying such clean equalitiesimmerses you into the use of reduction formulas,allowing for the simplification and evaluation of seemingly complex trigonometric integrals. Through reduction formulas, patterns emerge across problems, making them less intimidating and more approachable.
Mastering reduction formulas can greatly enhance your understanding and efficiency when solving integrals involving powers of sine and cosine functions. They provide a structured approach to systematically reducecomplex expressions down to solvable forms.