Question

In Exercises \(15-34,\) find the area of the regions enclosed by the lines and curves. $$y=\sec ^{2} x, \quad y=\tan ^{2} x, \quad x=-\pi / 4, \quad x=\pi / 4$$

Short answer

The area enclosed by the curves \(y = \sec^{2}x\), \(y = \tan^{2}x\), and the lines \(x = -\pi/4\) and \(x = \pi/4\) can be determined by the integral \(\int_{-\pi/4}^{\pi/4} (\sec^{2}x - \tan^{2}x) dx\). After solving it out, we will get the area. Due to the complexity of the integral, it might need to be solved using numerical methods or approximations.

Step-by-step solution

Sketch the Curves and Lines

Draw the graph of the two curves \(y = \sec^{2}x\) and \(y = \tan^{2}x\), and the lines \(x = -\pi/4\) and \(x = \pi/4\). This gives a visual representation of the area that needs to be determined.

Arrange the Functions

Determine which function is greater in the interval \([- \pi/4, \pi/4]\), which is \(y = \sec^{2}x\), and which one is less, \(y = \tan^{2}x\). This is essential for the later subtraction in the integral.

Set Up the Integral

Establish the integral for finding the area between the two curves over the range using the formula \(\int_{a}^{b} (f(x) - g(x)) dx\). Here, \(a\) and \(b\) are the points of intersection of the two curves, \(f(x) = \sec^{2}x\) and \(g(x) = \tan^{2}x\). So we have \(\int_{-\pi/4}^{\pi/4} (\sec^{2}x - \tan^{2}x) dx\).

Calculate the Integral

Solve the integral \(\int_{-\pi/4}^{\pi/4} (\sec^{2}x - \tan^{2}x) dx\). This will provide the area enclosed by the curves and lines.

Key concepts

Definite Integral

The concept of a definite integral is at the heart of calculating areas between curves. In essence, it represents the accumulation of quantities, which in geometric terms, corresponds to area under a curve between two specified points. To compute a definite integral, we apply the fundamental theorem of calculus, which establishes the relationship between differentiation and integration.

The process involves finding the antiderivative of the function being integrated, then evaluating this antiderivative at the upper and lower limits of integration, and finally taking the difference of these values. In terms of our exercise, the definite integral was used to calculate the area between the curves \(y = \text{sec}^2x\) and \(y = \text{tan}^2x\) over the interval \([-\pi/4, \pi/4]\).

Trigonometric Functions

In integral calculus, trigonometric functions often play a crucial role due to their periodic nature and applications in modeling oscillatory behavior. Functions such as sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent) repeat their values in a predictable manner, making them essential in both theoretical and practical contexts.

In our exercise example, the secant squared \(\sec^2x\) and tangent squared \(\tan^2x\) functions are central. It is important to understand how these functions behave, how they are related (\(\sec^2x = 1 + \tan^2x\)), and how they can be integrated. When working with these functions, students should be comfortable with their basic identities, as these are often used to simplify the integrals.

Integral Calculus

Integral calculus is a branch of mathematics focused on the processes of integration and their applications. It is the inverse of differential calculus, which deals with rates of change and slopes of curves. Integration allows us to find quantities like areas, volumes, and accumulations based on known functions.

To efficiently utilize integral calculus to solve real-world problems, such as finding the area between curves, one must understand the methods of integration, recognize function behavior, and determine the proper set-up of an integral formula. In the context of our session's problem, integral calculus is employed to find the area between the two curves by integrating the difference of the functions within the given limits.

Graphical Representation

A graphical representation is a visual way to display mathematical functions and can be exceedingly helpful to visualize the concept of area between curves. A graph provides a pictorial view of how functions behave, how they intersect, and where one might need to compute areas or distances.

In the case of the given exercise, sketching the curves \(y = \sec^{2}x\) and \(y = \tan^{2}x\), along with the vertical lines at \(x = -\pi/4\) and \(x = \pi/4\), gives a clear image of the region whose area we seek. This step is vital because it aids in understanding the physical meaning of the definite integral we compute and ensures the correct application of the integration process.