Problem 24 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(21-26\) $$ \int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( 1\mathbf{i} + \ln(2)\mathbf{j} + \frac{3}{4}\mathbf{k} \).

Step 1: Understand the Vector Integral

The given integral is a vector integral, which means it involves integrating each component of the vector function separately. The expression inside the integral is \((\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}\). We will evaluate each of these components separately over the interval \([0, \frac{\pi}{3}]\).

Step 2: Integrate the \(\mathbf{i}\) Component

The \(\mathbf{i}\) component is \(\sec t \tan t\). Since \(\frac{d}{dt}(\sec t) = \sec t \tan t\), this is a straightforward integration:\[\int \sec t \tan t\, dt = \sec t + C\]Evaluating from 0 to \(\frac{\pi}{3}\):\[\sec t \bigg|_{0}^{\pi/3} = \sec \left(\frac{\pi}{3}\right) - \sec(0) = 2 - 1 = 1\]

Step 3: Integrate the \(\mathbf{j}\) Component

The \(\mathbf{j}\) component is \(\tan t\). The integral of \(\tan t\) is \(\ln |\sec t| + C\):\[\int \tan t\, dt = \ln |\sec t| + C\]Evaluating from 0 to \(\frac{\pi}{3}\):\[\left(\ln |\sec t| \right)\bigg|_{0}^{\pi/3} = \ln \left(\sec \frac{\pi}{3}\right) - \ln (\sec 0) = \ln(2) - \ln(1) = \ln(2)\]

Step 4: Integrate the \(\mathbf{k}\) Component

The \(\mathbf{k}\) component is \(2 \sin t \cos t\). This can be simplified using the double angle identity for sine, \(\sin(2t) = 2\sin t \cos t\):\[\int 2 \sin t \cos t\, dt = \int \sin(2t) \, dt = -\frac{1}{2}\cos(2t) + C\]Evaluating from 0 to \(\frac{\pi}{3}\):\[-\frac{1}{2}\cos(2t) \bigg|_{0}^{\pi/3} = \left(-\frac{1}{2}\cos\left(\frac{2\pi}{3}\right)\right) - \left(-\frac{1}{2}\cos(0)\right) = -\frac{1}{2}\left(-\frac{1}{2}\right) - \left(-\frac{1}{2}\right) = \frac{1}{4} + \frac{1}{2} = \frac{3}{4}\]

Step 5: Combine the Results

Now, combine the results from each component:\[\int_{0}^{\pi/3} (\sec t \tan t) \mathbf{i} + (\tan t) \mathbf{j} + (2 \sin t \cos t) \mathbf{k} \, dt = 1\mathbf{i} + \ln(2)\mathbf{j} + \frac{3}{4}\mathbf{k}\]

Key Concepts

Trigonometric Integrals

When it comes to calculating integrals involving trigonometric functions, understanding key identities and techniques is important. In our example, we deal with integrals such as \( \sec t \tan t \), \( \tan t \), and \( 2 \sin t \cos t \). To tackle these efficiently:

• The integral of \( \sec t \tan t \) simplifies quickly as it is the derivative of \( \sec t \), making the integration straightforward. This is a handy result to remember.
• For \( \tan t \), one useful technique is to express it in terms of logarithmic functions, which is \( \ln |\sec t| + C \).
• When integrating \( 2 \sin t \cos t \), use the double angle identity \( \sin(2t) = 2 \sin t \cos t \). This reduces the problem to integrating \( \sin(2t) \).

Understanding these identities and their derivatives makes handling trigonometric integrals far simpler. Couple these with consistent practice to build a solid intuition around them.

Vector Calculus

Vector calculus unifies vectors and calculus, allowing the integration of vector-valued functions. In our exercise, the goal is to integrate each vector component separately because the vector function can be broken down into individual integral problems.Here's how you do it:

• Separate the vector into its components. Our vector is \((\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}\).
• Each component integrates individually over the same interval. This means that when you find the integral of one component, it does not affect the others. You simply add them back together to form a vector.

This separation and recombination are pivotal in vector calculus as they simplify complex vector operations into more manageable scalar operations.

Integration Techniques

There are various integration techniques to approach different kinds of integrals efficiently. Our solution involves techniques that simplify the integration process across different components.Some key techniques in our example include:

• **Substitution**: Useful when faced with trigonometric identities, as seen with \( 2 \sin t \cos t \). Recognizing the double angle formula helps simplify the integral.
• **Recognition of Derivatives**: Recognize derivations such as \( \frac{d}{dt}(\sec t) = \sec t \tan t \). This turns challenging integrals into straightforward results.
• **Logarithmic Integration**: Used in the integral of \( \tan t \), where its integral becomes a natural log function. This is essential for dealing with functions leading to infinite values or undefined behavior.

All these techniques can greatly reduce the complexity of an integral. Mastering these will enhance your ability to solve a broad range of problems in calculus.