Problem 13 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{\pi / 2}^{0} \frac{1+\cos 2 t}{2} d t$$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{\pi}{4}.\)

Step 1: Simplify the Integrand

The integrand is \( \frac{1 + \cos 2t}{2} \). This expression can be rewritten using the identity \( 1 + \cos 2t = 2 \cos^2 t \). Thus, the integrand becomes \( \frac{2 \cos^2 t}{2} = \cos^2 t \).

Step 2: Change the Limits of Integration

The integral has limits from \( t = \frac{\pi}{2} \) to \( t = 0 \). Changing the integration order requires a sign change, hence, \( \int_{\pi / 2}^{0} \cos^2 t \, dt = -\int_{0}^{\pi / 2} \cos^2 t \, dt \).

Step 3: Use Trigonometric Identity for Integration

Use the half-angle identity to further simplify: \( \cos^2 t = \frac{1 + \cos 2t}{2} \). Therefore, the integral becomes \( -\int_{0}^{\pi / 2} \frac{1 + \cos 2t}{2} \, dt \).

Step 4: Integrate Each Term Separately

Split the integral: \( -\int_{0}^{\pi / 2} \frac{1}{2} \, dt - \int_{0}^{\pi / 2} \frac{\cos 2t}{2} \, dt \).The first term \( -\int_{0}^{\pi / 2} \frac{1}{2} \, dt = -\left[ \frac{1}{2} t \right]_{0}^{\pi / 2} = -\frac{\pi}{4} \).The second term is \( -\frac{1}{2} \int_{0}^{\pi / 2} \cos 2t \, dt = -\frac{1}{2} \left[ \frac{1}{2} \sin 2t \right]_{0}^{\pi / 2} = 0 \).

Step 5: Combine Results

Add the results from the integrals: \(-\frac{\pi}{4} + 0 = -\frac{\pi}{4}.\)

Key Concepts

Trigonometric Identities

In integral calculus, trigonometric identities play a vital role in simplifying expressions and solving integrals. One common identity used in this context is the double-angle identity. For example, the identity \(1 + \cos 2t = 2 \cos^2 t\) is employed to simplify integrals involving trigonometric functions. By rewriting \(1 + \cos 2t\) as \(2 \cos^2 t\), we transform a complex expression into a simpler form, \(\cos^2 t\), which is easier to integrate.

Another useful identity is the half-angle identity: \(\cos^2 t = \frac{1 + \cos 2t}{2}\). This particular identity helps break down quadratic trigonometric functions into linear terms, making the integration process more straightforward. Identifying and correctly applying these identities can turn complex integral problems into manageable tasks, allowing us to focus on solving the integral efficiently.

Integration Techniques

Integration can often be tricky, especially with trigonometric functions, but employing the right techniques simplifies the process. For the given integral, the problem initially involves substitution by using known trigonometric identities to transform the integrand into a simpler form, like we did with \(\cos^2 t\). Once this simplification step is completed, one can proceed by breaking the integral into separate parts.

Each part of the integral can be handled individually; the integral of simple constants and linear trigonometric terms are often direct and intuitive. Additionally, changing the limits of integration when reversing their order requires incorporating a negative sign, as highlighted in the original solution. By using standard integration formulas for each separate term, such as \(\int \cos 2t \, dt = \frac{1}{2} \sin 2t + C\), we achieve the correct solution.

The strategy of decomposing complex integrals into smaller parts not only simplifies the process but also avoids potential mistakes when handling integral boundaries and identities.

Definite Integrals

Definite integrals represent the area under a curve between two distinct points on the x-axis. In this context, definite integrals involve limits of integration which dictate the boundary of the area being evaluated. Understanding how to work with these boundaries is crucial for solving these types of integrals accurately.

When calculating definite integrals, one must be careful with the limits, especially when the order of these limits is switched, as the integral changes sign. It is also important to remember that the result of a definite integral is a number and represents a net area, which could be positive, negative, or zero based on the function's placement relative to the x-axis.

In our example, calculating \(-\int_{0}^{\pi / 2} \cos^2 t \, dt\) involved evaluating the expression separately for each part of the integrand, yielding distinct numerical outcomes before combination, which resulted in \(-\frac{\pi}{4}\). Thus, mastery of definite integrals includes understanding how to handle the expressions both within the integrand and at their boundary values.