Problem 19 - Thomas Calculus

Question

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{\pi} 5(5-4 \cos t)^{1 / 4} \sin t d t $$

Step-by-Step Solution

Answer

The value of the integral is approximately 14.59.

Step 1: Identify the Substitution

To simplify the integration, we use substitution. Let \( u = 5 - 4 \cos t \). Then the derivative of \( u \) with respect to \( t \) is \( \frac{du}{dt} = 4 \sin t \).

Step 2: Solve for \( dt \)

Rewriting \( dt \) in terms of \( du \), we have \( du = 4 \sin t \, dt \). So, \( dt = \frac{du}{4 \sin t} \).

Step 3: Substitute in the Integral

Replace \( 5 - 4 \cos t \) with \( u \) and \( dt \) with \( \frac{du}{4 \sin t} \) in the integral: \[ \int_{0}^{\pi} 5(5-4 \cos t)^{1 / 4} \sin t \, dt = \int 5u^{1/4} \cdot \frac{du}{4} \] However, note that since \( \sin t \cdot \frac{1}{4 \sin t} \) cancels, making it:\[ \int \frac{5}{4} u^{1/4} du \]

Step 4: Change Limits of Integration

The limits for \( u \) need to be converted based on the substitution. When \( t = 0 \), \( u = 5 - 4 \times 1 = 1 \). When \( t = \pi \), \( u = 5 - 4 \times (-1) = 9 \). So, the new limits are from 1 to 9.

Step 5: Integrate the Substituted Function

With the limits updated, the integral becomes: \[ \int_{1}^{9} \frac{5}{4} u^{1/4} du \] This evaluates to:\[ \frac{5}{4} \cdot \left[ \frac{u^{5/4}}{5/4} \right]_{1}^{9} = \left[ u^{5/4} \right]_{1}^{9} \]

Step 6: Evaluate the Definite Integral

Evaluate \( \left[ u^{5/4} \right]_{1}^{9} \), which is:\[ 9^{5/4} - 1^{5/4} = 9^{5/4} - 1 \] Using a calculator, \( 9^{5/4} \approx 15.59 \). Thus, the result is:\[ 15.59 - 1 = 14.59 \]

Step 7: Finalize the Answer

Therefore, the integral evaluated is approximately 14.59.

Key Concepts

Definite Integrals

A definite integral allows us to calculate the net area under a curve between two specific points, known as limits of integration. In the given problem, the use of definite integration provides the net result of the function \( 5(5-4 \cos t)^{1 / 4} \sin t \) from \( t = 0 \) to \( t = \pi \).

• Definite integrals evaluate to a specific number rather than a function.
• The limits of 0 and \( \pi \) are placed at the bottom and top of the integral symbol, denoting the evaluation span.
• One crucial aspect of definite integration is that it considers not only the shape of the function but also specific points to provide a meaningful scalar result.

This process also involves calculating the area while considering the possibility of the function dipping below the x-axis (which would contribute negative area) and above the x-axis.

Change of Variables

The substitution method involves changing variables to simplify the integration process. In this exercise, the variable substitution \( u = 5 - 4 \cos t \) is made, simplifying the initial integration problem.

• Changing variables transforms complicated functions into simpler, more manageable forms.
• We also have to adjust the differential \( dt \) accordingly, which is solved here by expressing \( dt \) in terms of \( du \).
• This leads to replacing all parts of the original integral with their corresponding \( u \) terms, except for the limits.

Once the variable change is successfully applied, it is important to adjust the limits of integration to match the new variable. This is shown with converting \( t = 0 \) and \( t = \pi \) into \( u = 1 \) and \( u = 9 \), respectively. This ensures consistency throughout the integration.

Integration Techniques

There are numerous techniques to solve integrals, and substitution is one particularly powerful method for certain problems. This specific problem illustrates how substitution can simplify both indefinite and definite integrals.

• Initially, substitution aids in handling complicated expressions by reducing them to simpler ones.
• Following substitution, evaluation becomes straightforward, often reducing to elementary integrals.
• Substitution is especially beneficial when faced with integrands that involve compositions, like \( (5-4 \cos t)^{1/4} \), which can be challenging otherwise.

This technique involves systematically replacing complex parts of the integral with simpler terms and then modifying the limits accordingly, leading to a simpler form that can be integrated using basic rules. Mastering these techniques provides a solid foundation for tackling more complex integrals in calculus.