Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \sin ^{2} 3 x \cos 3 x d x $$

Step-by-Step Solution

Answer

\( \int_{0}^{\pi/2} \sin^2 3x \cos 3x \, dx = \frac{1}{9} \).

Step 1: Identify the Substitution

Let's use the substitution method for \ \( u = \sin 3x \). \ Then calculate the differential: \ \( du = 3\cos 3x \, dx \). \ This implies \ \( \cos 3x \, dx = \frac{1}{3} \, du \).

Step 2: Change the Limits of Integration

With our substitution \( u = \sin 3x \), we need to adjust the limits when \( x = 0 \) and \( x = \pi/2 \):\ - When \( x = 0 \), \( u = \sin 0 = 0 \).\ - When \( x = \pi/2 \), \( u = \sin (3 \times \pi/2) = \sin \frac{3\pi}{2} = -1 \).

Step 3: Substitute and Transform the Integral

Using the substitution and the new limits, the integral becomes: \ \[ \int_{0}^{-1} u^2 \left( \frac{1}{3} \, du \right) = \frac{1}{3} \int_{0}^{-1} u^2 \, du \].

Step 4: Evaluate the Integral in Terms of \( u \)

Compute the integral: \ \[ \frac{1}{3} \int u^2 \, du = \frac{1}{3} \left[ \frac{u^3}{3} \right]_{0}^{-1} = \frac{1}{3} \cdot \frac{1}{3} \left[ u^3 \right]_{0}^{-1} \].Evaluating gives: \[ \frac{1}{9} [ (-1)^3 - (0)^3 ] = \frac{1}{9} [ -1 - 0 ] = -\frac{1}{9} \].

Step 5: Interpret the Result Considering Direction

Since we originally integrated from zero to a negative value, our interpretation flips: \[ \int_{-1}^{0} u^2 \, du = \frac{1}{9} \].

Key Concepts

Substitution Rule

The Substitution Rule is a powerful technique used to solve definite integrals, especially when an integral involves a composition of functions. The idea is to simplify the integration process by transforming the variable of integration.
In the given exercise, the substitution is made by setting \( u = \sin 3x \). This choice stems from the trigonometric identity \( \sin 3x \cos 3x \) in the integral, which suggests potential simplification. After defining \( u \), you must find the differential \( du \). This gives \( du = 3\cos 3x \, dx \), which rearranges to \( \cos 3x \, dx = \frac{1}{3} \, du \).

• Choose the Substitution: Pick \( u \) such that it simplifies the integral.
• Find \( du \): Differentiate your substitution expression to relate \( du \) to \( dx \).
• Replace and Adjust Limits: Substitute \( u \) into the integral and adjust limits accordingly.

When applying these steps, the original bounds must be adjusted to reflect changes in the variable from \( x \) to \( u \). This creates an equivalent integral that's simpler to evaluate.

Calculus

Calculus is the mathematical study of continuous change, fundamentally focused on two main branches: differential calculus and integral calculus. For definite integrals, like in this exercise, it's part of integral calculus, which deals with the accumulation of quantities and the area under and between curves.
Definite integrals offer precise numerical results and are bounded by lower and upper limits of integration. Here, limits \( x = 0 \) to \( x = \frac{\pi}{2} \) are transformed to \( u = 0 \) to \( u = -1 \) because of substitution. This transformation reflects the change in variables, maintaining consistency in the integration range.

• Understanding Limits: These define the range over which you integrate.
• Accurate Seamless Transformation: When changing variables, ensure the limits adapt accordingly.
• Fundamental Theorem of Calculus: Connects differentiation with integration, facilitating the evaluation.

Calculus provides the foundation for solving this problem by ensuring transformations between variables preserve the analytical structure of integrals.

Integration Techniques

Integration techniques are various methods used to evaluate integrals, each with its special suitability depending on the form of the function. Here, a combination of substitution and simplification techniques allows the integral \( \int_{0}^{\pi / 2} \sin ^{2} 3 x \cos 3 x \, dx \) to be evaluated.
The integration process includes:

• Substitution Simplification: Utilizing \( u = \sin 3x \) simplifies the expression, transforming \( \sin^2 3x \cos 3x \) into \( u^2 \).
• Integral Evaluation: After substitution, integrate \( u^2 \) using power rule, which results in \( \frac{u^3}{3} \).
• Definite Integral Evaluation: Substituting limits gives a final value, adjusted for any directional interpretation needed depending on the bounds.

Such integration techniques, particularly when combined with substitution and adjustment of bounds, effectively solve complex integrals by reducing them to more manageable forms and interpreting results correctly within physical or mathematical contexts.