Problem 56 - Calculus Early Transcendentals: Pearson New International Edition

Question

Integrals that occur frequently in applications are \(\int_{0}^{2 \pi} \cos ^{2} x d x\) and \(\int_{0}^{2 \pi} \sin ^{2} x d x\) (a) Using a trigonometric identity, show that $$ \int_{0}^{2 \pi}\left(\sin ^{2} x+\cos ^{2} x\right) d x=2 \pi $$ (b) Show from graphical considerations that $$ \int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x $$ (c) Conclude that \(\int_{0}^{2 \pi} \cos ^{2} x d x=\int_{0}^{2 \pi} \sin ^{2} x d x=\pi\).

Step-by-Step Solution

Answer

\(\int_0^{2\pi} (\sin^2 x + \cos^2 x) \, dx = 2\pi\), \(\int_0^{2\pi} \cos^2 x \, dx = \int_0^{2\pi} \sin^2 x \, dx = \pi\).

Step 1: Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). This identity is fundamental in trigonometry and simplifies the integration.

Step 2: Integrate the Identity over the Interval

Integrate both sides of the identity over \([0, 2\pi] \):\[\int_0^{2\pi} (\sin^2 x + \cos^2 x) \, dx = \int_0^{2\pi} 1 \, dx\].

Step 3: Calculate the Integral

The integral of 1 with respect to \(x\) over the interval \([0, 2\pi]\) is simply the length of the interval:\[\int_0^{2\pi} 1 \, dx = 2\pi\].Thus, \(\int_0^{2\pi} (\sin^2 x + \cos^2 x) \, dx = 2\pi\), showing part (a).

Step 4: Graphical Considerations for Equality

The functions \(\cos^2 x\) and \(\sin^2 x\) have the same graph shape, just phase-shifted by \(\frac{\pi}{2}\) units. Over \([0, 2\pi]\), each function covers a full cycle. This symmetry implies\[\int_0^{2\pi} \cos^2 x \, dx = \int_0^{2\pi} \sin^2 x \, dx\].This concludes part (b).

Step 5: Conclusion Using Previous Results

We know \(\int_0^{2\pi} (\sin^2 x + \cos^2 x) \, dx = 2\pi\) and from symmetry\(\int_0^{2\pi} \cos^2 x \, dx = \int_0^{2\pi} \sin^2 x \, dx\). Let these values be \(I\). Then\[2I = 2\pi \I = \pi\].Thus, \(\int_0^{2\pi} \cos^2 x \, dx = \int_0^{2\pi} \sin^2 x \, dx = \pi\), completing part (c).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric equation expressed as \( \sin^2 x + \cos^2 x = 1 \). This identity highlights the inherent relationship between sine and cosine functions. It is named after the Pythagorean Theorem due to its resemblance in expressing a relation similar to that of sides in a right triangle.

In the context of integration, using this identity allows simplification of expressions involving squares of sine and cosine. For example, integrating \( \sin^2 x + \cos^2 x \) over a specific interval can be reduced to integrating the constant function 1. This simplifies the computation significantly, especially over the interval \([0, 2\pi]\). Understanding and applying this identity is crucial in solving many trigonometric integrals.

Integration Techniques

Integration techniques involve methods to find the antiderivative of functions. In trigonometric integrals, strategies often include applying known identities to simplify expressions before integrating.

• Substitution and Transformation: Substantial simplification occurs by transforming the problem using identities like the Pythagorean Identity.
• Direct Integration: For constants and simple linear functions, direct integration is straightforward and efficient.
• Splitting Intervals: Integrals over symmetric intervals can be shared between parts, easing the computation.

In our exercise, we used the trigonometric identity to integrate \( 1 \) directly over the interval \([0, 2\pi]\), leading to an easy calculation of \(2\pi\).

Graphical Symmetry

The concept of graphical symmetry plays a vital role in trigonometric integrations. Functions like \( \cos^2 x \) and \( \sin^2 x \) exhibit symmetry over a full cycle. This symmetry helps intuitively boost understanding.

Both functions \( \cos^2 x \) and \( \sin^2 x \) appear consistently as identical wave patterns, just shifted horizontall by \(\frac{\pi}{2}\). This shift implies that despite appearing different in equations, they share equal integrals when considered over the entire interval \([0, 2\pi]\). Because they cover a full cycle within the interval, their areas under the curve are equivalent. Thus, this symmetry underpins equations part (b) and confirms equal integration results for both functions.

Definite Integrals

Definite integrals refer to the integration of a function within specified limits or intervals. It not only calculates the area under a curve but also delineates the net area over an interval.

• Value Determination: For \([0, 2\pi]\), the definite integral gives the exact total area.
• Applications: Used in physics to calculate displacement, areas, and other physical quantities.

In our problem, the integration of \( \cos^2 x + \sin^2 x \) over \([0, 2\pi]\) not only assures the identity holds but confirms calculations mathematically solidifying our concluding results. Thus, solving it demonstrated how definite integrals encapsulate function behavior over an interval, leading to meaningful conclusions like proving areas of \( \cos^2 x \) and \( \sin^2 x \) are equal to \( \pi \).