Problem 89 - Precalculus

Question

In calculus, the definite integral \(\int_{a}^{b} f(x) d x\) is used to find the area below the graph of a continuous function \(f\), above the \(x\) -axis, and between \(x=a\) and \(x=b\). The Fundamental Theorem of Calculus cstablishes that the definite integral \(f_{a}^{b} f(x) d x\) equals \(F(b)-F(a),\) where \(F\) is any antiderivative of a continuous function \(f.\) In Exercises \(87-90\), first shade the area corresponding to the definite integral and then use the information below to find the exact value of the area. $$\begin{array}{|l|c|c|c|c|} \hline \text { Function } & \tan x & \cot x & \sec x & \csc x \\ \hline \text { Antiderivative } & -\ln |\cos x| & \ln |\sin x| & \ln |\sec x+\tan x| & -\ln |\csc x+\cot x| \\ \hline \end{array}$$ $$\int_{0}^{\pi / 4} \sec x d x$$

Step-by-Step Solution

Answer

The exact value of the area is \( \ln(\sqrt{2} + 1) \).

Step 1: Identify the function and limits of integration

We are given the integral \( \int_{0}^{\pi/4} \sec x \, dx \). This is a definite integral where the function is \( \sec x \) and the limits of integration are from \( x = 0 \) to \( x = \pi/4 \).

Step 2: Find the antiderivative of the function

According to the table provided, the antiderivative of \( \sec x \) is \( \ln | \sec x + \tan x | \). This means that if we differentiate \( \ln | \sec x + \tan x | \), we will get \( \sec x \).

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral of a function \( f(x) \) from \( a \) to \( b \) is equal to \( F(b) - F(a) \), where \( F \) is an antiderivative of \( f \). Here, the integral \( \int_{0}^{\pi/4} \sec x \, dx \) can be evaluated as \( F(\pi/4) - F(0) \).

Step 4: Evaluate the antiderivative at the upper limit

Calculate \( F(\pi/4) \) using the antiderivative: \( \ln | \sec(\pi/4) + \tan(\pi/4) | \). Since \( \sec(\pi/4) = \sqrt{2} \) and \( \tan(\pi/4) = 1 \), this becomes \( \ln | \sqrt{2} + 1 | \).

Step 5: Evaluate the antiderivative at the lower limit

Calculate \( F(0) \) using the antiderivative: \( \ln | \sec(0) + \tan(0) | \). Since \( \sec(0) = 1 \) and \( \tan(0) = 0 \), this simplifies to \( \ln | 1 + 0 | = \ln(1) = 0 \).

Step 6: Calculate the definite integral

Substitute the values from the previous steps into the Fundamental Theorem of Calculus: \( F(\pi/4) - F(0) = \ln | \sqrt{2} + 1 | - 0 = \ln(\sqrt{2} + 1) \).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a central principle in calculus that connects differentiation and integration. It is essential for evaluating definite integrals. This theorem states that if a function is continuous over a certain interval and you have an antiderivative of this function, then the definite integral over this interval can be calculated using the values of the antiderivative at the endpoints.

In simplest terms, if you have a function \( f(x) \) and its antiderivative \( F(x) \), the definite integral from \( a \) to \( b \) can be written as \( F(b) - F(a) \). This means you only need to find the antiderivative, evaluate it at the endpoint values, and subtract the result for the lower limit from the upper limit result.

In the exercise, we used the Fundamental Theorem of Calculus to evaluate \( \int_{0}^{\pi/4} \sec x \, dx \). By identifying \( F(x) \) from the problem, we computed the integral as \( \ln(\sqrt{2} + 1) \). This process simplifies the integration of complex functions and highlights the beauty of calculus for solving real-world problems.

Antiderivatives

Antiderivatives are the reverse of derivatives, and they are fundamental in solving integrals. Calculating an antiderivative is often referred to as finding the 'indefinite integral'. An antiderivative of a function \( f(x) \) is another function \( F(x) \) such that the derivative of \( F(x) \) returns \( f(x) \).

In our exercise, the task was to find the antiderivative of \( \sec x \). According to the provided table, the antiderivative is given by \( \ln | \sec x + \tan x | \). To solve the definite integral \( \int_{0}^{\pi/4} \sec x \, dx \), we had to correctly identify this antiderivative.

Understanding how to find antiderivatives is crucial for calculating integrals, mainly because the Major Theorem of Calculus depends on these expressions for evaluating areas under curves. Always ensure the correctness of the antiderivative because any mistakes here might lead to incorrect results in the integral calculations.

Trigonometric Functions

Trigonometric functions like sine, cosine, tangent, secant, are periodic functions that arise often in calculus problems, especially integrals. They describe properties of triangles and are essential in modeling cyclical phenomena like waves or oscillations.

In the context of integration, these functions can be tricky due to their periodic nature. Secant, for example, is defined as \( \sec x = \frac{1}{\cos x} \) and has its peculiarity in calculations. When integrating functions like \( \sec x \), it's often beneficial to know their antiderivatives, which can be found in specialized tables or through specific integral calculus techniques.

In the exercise, the integral of interest involved \( \sec x \) between specific limits. By leveraging known antiderivatives of trigonometric functions, which provide a direct way to compute definite integrals, the process was simplified. Hence, familiarity with these functions and their integrals will greatly aid in tackling various calculus problems effectively.