Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta$$

Short answer

Answer: The value of the given definite integral is 1.

Step-by-step solution

Identify the function and its antiderivative

Our function is: $$f(\theta) = \csc ^{2} \theta$$ The antiderivative of \(\csc^2{\theta}\) is also known as its indefinite integral. The indefinite integral of \(\csc ^{2} \theta\) is given by: $$F(\theta) = -\cot \theta + C$$ Here, C is the constant of integration which we don't need to worry about since we are dealing with definite integrals.

Apply Fundamental Theorem of Calculus

Now that we have found the antiderivative, we will apply the Fundamental Theorem of Calculus: $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta = F(\frac{\pi}{2}) - F(\frac{\pi}{4})$$ Substitute the antiderivative, $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta = [-\cot(\frac{\pi}{2})] - [-\cot(\frac{\pi}{4})]$$

Evaluate the antiderivative at the endpoints

Compute the values of cotangent at \(\pi/2\) and \(\pi/4\): $$\cot(\frac{\pi}{2}) = 0\ \text{and}\ \cot(\frac{\pi}{4}) = 1$$ Now substitute these values in the expression: $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta = [-0] - [-1] = 1$$ Thus, the value of the definite integral is: $$\int_{\pi / 4}^{\pi / 2} \csc ^{2} \theta d \theta = 1.$$

Key concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a fundamental connection between differentiation and integration, two core concepts in calculus. It serves as a powerful tool that allows us to evaluate definite integrals without the cumbersome task of approximating them. The theorem is composed of two parts:

• The first part states that if a function is continuous over a closed interval, then it has an antiderivative over that interval.
• The second part, which we use in this problem, states that the definite integral of a function over an interval can be determined using its antiderivative.

Here, we apply the second part. After finding the antiderivative of the function, we evaluate it at the upper and lower bounds of our interval, and subtract the results. This is the essence of the theorem, creating a bridge between the infinite process of finding areas under curves and simple arithmetic.

Antiderivative

An antiderivative, or indefinite integral, is a function whose derivative is the original function we started with. To visualize this concept, imagine working backward from finding a derivative. In our exercise, we sought the antiderivative of the cosecant squared function, \[\int \csc^2 \theta \ d\theta = -\cot\theta + C\]The constant \(C\) is usually omitted in definite integrals because it cancels out during subtraction.

• While derivatives give us the rate of change, antiderivatives help in finding accumulated values, like areas under curves.
• The process of finding antiderivatives is crucial for solving integrals effectively and this forms the backbone for calculating definite integrals.

Cosecant function

The cosecant function, denoted as \(\csc\theta\), is the reciprocal of the sine function. Therefore, \(\csc \theta = \frac{1}{\sin \theta}\). It is undefined wherever the sine function is zero because division by zero is not possible.Understanding the trigonometric identities is vital when working with integrals involving trigonometric functions. Here,

• The derivative of \(-\cot\theta\) results in \(\csc^2\theta\), connecting back to the original function whose integral we evaluated.
• Recognizing these relationships helps simplify integration of trigonometric functions.

Being familiar with these identities enhances mathematical fluency and understanding, making integration problems like these more approachable.

Definite integral evaluation

Evaluating definite integrals involves calculating the specific value of an integral over a defined range. This turns an abstract area-under-the-curve concept into a concrete number. Using the evaluation from our exercise: First, identify the antiderivative \(-\cot\theta\) and then apply it to the bounds: \[\int_{\pi/4}^{\pi/2} \csc^2 \theta \, d\theta = \left[ -\cot\left(\frac{\pi}{2}\right) \right] - \left[ -\cot\left(\frac{\pi}{4}\right) \right]\]Calculating the cotangent at these angles, we find:

• \(-\cot\left(\frac{\pi}{2}\right) = 0\)
• \(-\cot\left(\frac{\pi}{4}\right) = -1\)

Therefore, the final result of the definite integral is: \(0 - (-1) = 1\).

• This process showcases the beauty of calculus, turning infinite processes into finite computations.