Problem 31 - Essential Calculus

Question

In Exercises 31 and 32, use a computer algebra system to determine the antiderivative that passes through the given point. Use the system to graph the resulting antiderivative. $$ \int \frac{1}{\sin \theta \tan \theta} d \theta,\left(\frac{\pi}{4}, 2\right) $$

Step-by-Step Solution

Answer

The antiderivative that passes through the point \( (\frac{\pi}{4}, 2) \) is \( \sin \theta + 2 - \sqrt{2} \).

Step 1: Simplify the integrand

The given integral is quite complex, but it can be simplified by using a trigonometric identity: \(1 / \sin \theta \tan \theta = \cos \theta\). So the integral becomes \[ \int \cos \theta d \theta \], which is easier to compute.

Step 2: Compute the antiderivative

Now we find the antiderivative of \( \cos \theta \), which is \( \sin \theta \). But we need to add an arbitrary constant of integration, C, so finally, the antiderivative of the initial integral is: \[ \sin \theta + C \]

Step 3: Determine the constant of integration

We were given that the function passes through the point \((\frac{\pi}{4}, 2)\). This means, when \( \theta = \frac{\pi}{4} \), the function value is 2. So we substitute these values into the antiderivative to solve for C:\[ 2 = \sin \left(\frac{\pi}{4}\right) + C \]This implies that C is \( 2 - \sqrt{2} \). Hence, the specific antiderivative that passes through the given point is:\[ \sin \theta + 2 - \sqrt{2} \].

Step 4: Graph the resulting antiderivative

To complete the exercise, we would graph the antiderivative function. The horizontal axis represents the measure of the angle \( \theta \) in radians, and the vertical axis represents the function value. Without a visual representation provided, you would need to use your computer algebra system to draw this graph, but from known properties of the sine function you could expect a wave-like graph, shifted vertically upward by \( 2 - \sqrt{2} \) units.

Key Concepts

Integration

Integration is a fundamental tool in calculus that allows us to find the total accumulation of quantities such as area, volume, or mass. In the context of trigonometric functions, integration can be thought of as the process of finding the antiderivative or the inverse operation of differentiation. When you integrate a function like \( \cos \theta \), you're effectively summing up small changes to calculate a whole.

For many students, visualizing this process can help. Imagine tracing the graph of \( \cos \theta \), and integration would give you the area under this curve up to that point. The integral \( \int \cos \theta d \theta \) represents this accumulation, and the result is a function--in this case, \( \sin \theta \)--that expresses this cumulative total. Adding a constant of integration accounts for the fact that there are infinitely many antiderivatives, each differing by a constant amount.

Trigonometric Identities

Trigonometric identities are equations that relate the various trigonometric functions to one another. They're essential tools for simplifying expressions and solving problems involving triangles and periodic phenomena. The identity used in our example, \(1 / \sin \theta \tan \theta = \cos \theta\), falls under reciprocal identities, which express one trigonometric function in terms of the reciprocal of another.

To apply trigonometric identities effectively, it helps to have a strong grasp of the fundamental trig identities, such as Pythagorean identities, co-function identities, and even/odd identities. Knowing these allows for a variety of integration problems to be transformed into more workable forms, just as the complex-appearing integral in the exercise was reduced to a simple \( \int \cos \theta d \theta \).

Definite Integrals

Definite integrals provide specific numerical answers when we integrate between two limits, which tells us the exact accumulation of a quantity over a specific interval. In contrast to indefinite integrals, which give a general antiderivative function plus a constant, definite integrals represent the exact area under the curve between two points.

In situations where we aren't provided with limits, as is the case with our exercise, we deal with an indefinite integral and then determine the arbitrary constant of integration, represented by 'C', by using a point that the function must pass through. This specific adjustment, from the general to the particular, is what turns an indefinite integral into a function with well-defined conditions that model a real-world situation or meet particular requirements, such as going through the point \(\left(\frac{\pi}{4}, 2\right)\) in our example.

Computer Algebra System

A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics. These systems are incredibly valuable for solving complex algebraic structures, performing differentiation and integration, and even graphing functions. In educational settings, they help students visualize the results and understand the behavior of functions.

In the context of our exercise, a CAS would be used not only to verify the steps of an antiderivative calculation but also to plot the graph of the resulting function. This capability is particularly useful when the function involves trigonometric expressions, as it helps illustrate properties such as periodicity and amplitude. The ability to graph the adjusted antiderivative function--the \( \sin \theta \) term shifted vertically by \( 2 - \sqrt{2} \) units in this case--provides students with a visual understanding of how the function behaves and how it corresponds to the given point through which it passes.