Question

Show that \(\int_{0}^{\infty} \sin \theta \mathrm{d} \theta\) and \(\int_{0}^{\infty} \cos \theta \mathrm{d} \theta\) are indeterminate.

Short answer

Answer: Both integrals are indeterminate because when evaluating them, the limits do not converge to a finite value. The sine and cosine functions oscillate between -1 and 1 as the limit goes to infinity, which results in the integrals being indeterminate.

Step-by-step solution

Find the anti-derivative of sine function

To find the anti-derivative of sine function, recall that: $$ \frac{d}{d\theta}(-\cos \theta) = \sin \theta $$ So, the anti-derivative of sine function is: $$ \int \sin \theta \, d\theta = -\cos \theta + C $$

Evaluate the integral of sine function

Now, we will evaluate the integral of sine function: $$ \int_{0}^{\infty} \sin \theta \, d\theta = \lim_{T \to \infty} \int_{0}^{T} \sin \theta \, d\theta $$ Using the anti-derivative found in Step 1: $$ \lim_{T \to \infty} [-\cos \theta]_{0}^{T} = \lim_{T \to \infty} \left( -\cos T - (-\cos 0) \right) $$ As \(T\) goes to infinity, the cosine function oscillates between \(-1\) and \(1\). Therefore, the limit does not exist, and the integral is indeterminate.

Find the anti-derivative of cosine function

To find the anti-derivative of cosine function, recall that: $$ \frac{d}{d\theta}( \sin \theta) = \cos \theta $$ So, the anti-derivative of cosine function is: $$ \int \cos \theta \, d\theta = \sin \theta + C $$

Evaluate the integral of cosine function

Now, we will evaluate the integral of cosine function: $$ \int_{0}^{\infty} \cos \theta \, d\theta = \lim_{T \to \infty} \int_{0}^{T} \cos \theta \, d\theta $$ Using the anti-derivative found in Step 3: $$ \lim_{T \to \infty} [\sin \theta]_{0}^{T} = \lim_{T \to \infty} \left( \sin T - \sin 0 \right) $$ As \(T\) goes to infinity, the sine function oscillates between \(-1\) and \(1\). Therefore, the limit does not exist, and the integral is indeterminate. In conclusion, both integrals are indeterminate as their limits do not converge to a finite value.

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematical analysis dealing with the accumulation of quantities and the area under curves. When you are working with integrals, you are essentially looking for the total accumulation of a quantity, which can be represented in terms of area under the graph of a function. This concept is crucial for understanding how quantities evolve over an interval.

An integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. There are two main types of integrals: definite and indefinite. An indefinite integral, or anti-derivative, represents a family of functions whose derivative is the integrand. On the other hand, a definite integral computes the accumulation over a specific interval, which can be thought of as the 'net' area between the x-axis and the function's graph from one limit of integration to another.

Anti-derivative

The anti-derivative, also known as the primitive or indefinite integral, is the reverse process of differentiation. In simple terms, finding an anti-derivative means discovering a function whose rate of change (derivative) is equal to the given function.

For any continuous function, there are infinitely many anti-derivatives, each differing by a constant. For example, the anti-derivative of \(\theta\) is \(\frac{1}{2} \theta^2 + C\), where \(C\) is the constant of integration. The role of the anti-derivative in integral calculus is paramount as it provides the function needed to evaluate definite integrals through the Fundamental Theorem of Calculus.

Oscillating Functions

Oscillating functions, such as the sine and cosine functions, have values that continue to increase and decrease in a repeatable pattern. The sine and cosine functions are periodic, which means they repeat their values in regular intervals, known as the period of the function.

When dealing with the integrals of oscillating functions over an infinite interval, the behavior of the function can lead to an indeterminate result if it doesn't settle down to a single value. This is due to the fact that as the function continues to oscillate, the total accumulation over time doesn't converge to a finite number, which is precisely what we observe in the given exercise where the integrals of sine and cosine functions over an infinite interval are indeterminate.

Limits of Integration

The limits of integration define the interval over which the definite integral is computed. In other words, they specify the start and end points of the accumulation process. For finite intervals, the limits of integration are just the numbers that bound the interval. However, when evaluating integrals over infinite intervals, one employs limits to define the endpoints.

This is often done using the concept of a limit from calculus, where we let the upper limit of integration approach infinity. The notation \(\int_{a}^{\infty} f(x) dx\) literally means 'the limit of \(\int_{a}^{T} f(x) dx\) as T approaches infinity'. If the integral settles down to a finite value, we say the integral converges. If not, as is the case with the oscillating sine and cosine functions in the exercise, the integral is called divergent or indeterminate.