Problem 186 - Calculus Volume 1

Question

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 . $$ \int_{\pi / 3}^{\pi / 4} \csc \theta \cot \theta d \theta $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\sqrt{2} + \frac{2}{\sqrt{3}}\).

Step 1: Identify the Integral Components

To solve the integral \( \int_{\pi/3}^{\pi/4} \csc \theta \cot \theta \, d\theta \), first note the integrand \( \csc \theta \cot \theta \). This form suggests a function in terms of \( \theta \) that can be integrated easily using trigonometric identities or known derivatives.

Step 2: Find the Antiderivative

Recall that the derivative of \( -\csc \theta \) is \( \csc \theta \cot \theta \). Thus, the antiderivative of \( \csc \theta \cot \theta \) is \( -\csc \theta \).

Step 3: Apply the Fundamental Theorem of Calculus (Part 2)

Using the Fundamental Theorem of Calculus, Part 2, evaluate the integral by first substituting the bounds into the antiderivative. Calculate: \[-\csc \left( \frac{\pi}{4} \right) - \left(-\csc \left( \frac{\pi}{3} \right) \right).\]

Step 4: Calculate the Cosecant Values

Compute \( \csc \left( \frac{\pi}{4} \right) \) and \( \csc \left( \frac{\pi}{3} \right) \). Note that \( \csc \theta = \frac{1}{\sin \theta} \): - \( \csc \left( \frac{\pi}{4} \right) = \frac{1}{\sin \frac{\pi}{4}} = \sqrt{2} \). - \( \csc \left( \frac{\pi}{3} \right) = \frac{1}{\sin \frac{\pi}{3}} = \frac{2}{\sqrt{3}} \).

Step 5: Substitute and Simplify

Substitute these values back into the expression: \[-\sqrt{2} + \frac{2}{\sqrt{3}}.\]Then, simplify to find the exact result of the integral.

Step 6: Final Simplification

Combine the terms to simplify the expression. To combine \( -\sqrt{2} + \frac{2}{\sqrt{3}} \), find a common denominator or rationalize as necessary. This results in the simplified expression for the integral.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve, bounded by a specific interval on the x-axis. This kind of integral allows us to calculate the total accumulation or net change over an interval.
Definite integrals are expressed with limits of integration, which are the boundaries for the variable of integration.

• In the original exercise, these boundaries are \(\pi/3\) (lower bound) and \(\pi/4\) (upper bound).

By solving a definite integral, you're often finding the difference between the values of an antiderivative at these limits. In essence, a definite integral combines the process of integration and evaluation to provide a comprehensive result.
In practical applications, this process can be used to find areas, volumes, displacement, and more. It's a foundational tool in calculus that bridges analytic and geometric interpretations.

Antiderivatives

An antiderivative is the reverse process of differentiation. It represents a function whose derivative leads back to the original function you started with.
For example, if the derivative of a function \( f(x) \) is \( g(x) \), then an antiderivative of \( g(x) \) would be \( f(x) + C \), where \( C \) is the constant of integration.

• In our exercise, we see that the antiderivative of \( \csc \theta \cot \theta \) is \(-\csc \theta\).

The process of finding antiderivatives is crucial when you are solving a definite integral using the Fundamental Theorem of Calculus.
Understanding antiderivatives allows you to manipulate and solve integral problems more effectively. It's a critical skill that forms the basis for solving more complex calculus problems. By recognizing these patterns, you can simplify and tackle various types of integrals.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable. They are essential for simplifying expressions and solving integrals or equations that involve trigonometric functions.
Some common identities include:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

In the solution provided, the identity \( \csc \theta = \frac{1}{\sin \theta} \) is used to evaluate cosecant values when determining the definite integral's value.
These identities help break down complex expressions into simpler forms, making calculations more manageable. They are vital tools in calculus for working with trigonometric functions within different kinds of mathematical problems.