Problem 25 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find the indicated integral. $$\int_{0}^{\pi} \cos \left(\frac{x}{3}\right) d x$$

Step-by-Step Solution

Answer

\( \frac{3\sqrt{3}}{2} \)

Step 1: Set up the integral

Set up the integral as given: \[ I = \int_{0}^{\pi} \cos \left( \frac{x}{3} \right) dx \]

Step 2: Make a substitution

To simplify, use the substitution: \[ u = \frac{x}{3} \] Then, differentiate both sides: \[ du = \frac{1}{3} dx \] So, \[ dx = 3 du \]

Step 3: Change the limits of integration

Change the limits of integration to match the substitution: When \( x = 0 \), \( u = 0 \) When \( x = \pi \), \( u = \frac{\pi}{3} \)

Step 4: Rewrite the integral with the substitution

Replace \( x \) and \( dx \) in the integral: \[ I = \int_{0}^{\frac{\pi}{3}} \cos(u) \( 3 du \) \]

Step 5: Simplify the integral

Factor out the constant 3: \[ I = 3 \int_{0}^{\frac{\pi}{3}} \cos(u) du \]

Step 6: Integrate

Integrate \(\cos(u)\): \[ \int \cos(u) du = \sin(u) \] So the integral becomes: \[ I = 3 [\sin(u)]_{0}^{\frac{\pi}{3}} \]

Step 7: Evaluate the definite integral

Evaluate \(\sin(u)\) at the limits: \[ I = 3 ( \sin(\frac{\pi}{3}) - \sin(0) ) \] \[ \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \] and \( \sin(0) = 0 \)

Step 8: Simplify the result

Substitute back these values: \[ I = 3 ( \frac{\sqrt{3}}{2} - 0 ) = \frac{3\sqrt{3}}{2} \]

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve between two specified points.
A definite integral is written as \(\int_{a}^{b} f(x) dx\) where \(a\) and \(b\) are the limits of integration. In the given exercise, these limits are from \(0\) to \(\pi\).
To solve a definite integral, we follow these general steps:

• Set up the integral based on the given function and limits.
• Simplify the integral using techniques like substitution.
• Evaluate the resulting indefinite integral.
• Lastly, apply the limits to find the final result.

A definite integral has a geometric interpretation: it measures the area under the curve of the function within the limits. Positive areas are above the x-axis, while negative areas are below it. This dual nature captures a complete picture of how the function behaves between the specified limits.
Understanding definite integrals helps in many fields, such as physics for calculating the total displacement given a velocity function, or in economics for finding the area under a supply or demand curve.

Trigonometric Integration

Integrating trigonometric functions like \(\text{cos}(x)\) and \(\text{sin}(x) \) is common in calculus.
Trigonometric integrals involve integrating functions involving trigonometric expressions. In our example, we integrate \(\text{cos} \left(\frac{x}{3}\right)\). Key steps for trigonometric integration often include:

• Using standard integral formulas of trigonometric functions. For instance, \(\int \text{cos}(u) \ du = \sin(u)\)
• Simplifying using trigonometric identities or substitutions, to make integration easier.

In the example, we used substitution to simplify \(\text{cos} \left(\frac{x}{3}\right)\) into a basic \(\text{cos}(u)\) function, which is straightforward to integrate. Remember:

• \text{sin}(u)' = \text{cos}(u)
• \text{cos}(u)' = -\text{sin}(u)

Mastering trigonometric integration helps with handling problems in physics, engineering, and other areas where wave forms and oscillatory functions are prevalent.

Substitution Method

The substitution method is a powerful tool for simplifying integrals. Here’s how it works:
We choose a substitution \(u = g(x)\) that simplifies the given integral. For our problem:

• The substitution \(u = \frac{x}{3} \) helped simplify \(\text{cos} \left(\frac{x}{3}\right)\) to \(\text{cos}(u)\).

The rationale for using substitution includes:

• Making the integrand easier to manage.
• Transforming a complex function into a simpler one.

Steps for Substitution:

1. Choose a substitution \( u = g(x) \).
2. Differentiate to find \( du \).
3. Replace all instances of the original variable \((x) \) and \(dx \) in the integral.
4. Rewrite the limits of integration, if it's a definite integral.
5. Integrate with respect to the new variable \( u \) and then revert back to the original variable.

The substitution method is essential for transforming difficult integrals into forms where standard integral formulas apply. It’s widely used in various calculus problems to achieve easier computations and exact solutions.