Question

Evaluate the integral \(\int_{0}^{\pi / 6} \sin ^{2} x d x\) by doing the following: a. Compute the integral exactly. b. Integrate the first three terms of the Maclaurin series expansion of the integrand and compare with the exact result.

Short answer

The exact integral can be calculated using trigonometric identities while the approximate integral is found by integrating the first three terms of the Maclaurin series expansion. A comparison of these results will reveal the accuracy of the Maclaurin series approximation.

Step-by-step solution

Compute the integral exactly

The integral \(\int_{0}^{\pi / 6} \sin^{2}(x) dx\) involves a trigonometric function. First, it's important to express \(\sin^{2}(x)\) in terms of cosine using the identity \(\sin^{2}(x)=\frac{1-\cos (2 x)}{2}\). Then, we integrate from 0 to \(\pi/6\) to obtain the exact result.

Find the Maclaurin series expansion

The Maclaurin series expansion for \(\sin(x)\) is \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\). The square of this sequence gives the expansion for \(\sin^{2}(x)\). Collect the first three terms of this expansion.

Integrate the first three terms of the Maclaurin series

Next, the first three terms of the Maclaurin series expansion of \(\sin^{2}(x)\) will be integrated from 0 to \(\pi/6\).

Compare results

Once the integrals are calculated, both results should be compared to see how accurate the Maclaurin series approximation is in relation to the exact result.

Key concepts

Trigonometric Integrals

Trigonometric integrals are integrals of functions that include trigonometric functions like sine, cosine, and tangent. These types of integrals often appear in many areas of mathematics, physics, and engineering. When dealing with a trigonometric integral like \(\int \sin^2(x) \, dx\), one helpful technique is to use trigonometric identities to simplify the expression.
For example, the trigonometric identity \(\sin^2(x) = \frac{1-\cos(2x)}{2}\) can be very useful. By converting \(\sin^2(x)\) into this expression, the integral becomes much easier to solve. The integral \(\int \cos(2x) \, dx\) is simpler and more standard, enabling straightforward calculation.

• Break down complex functions.
• Utilize identities to simplify the integration process.
• Calculate the integral over the given bounds, such as from 0 to \(\pi/6\).

This approach is not only efficient but also ensures that you achieve an exact result when working with trigonometric integrals.

Maclaurin Series

The Maclaurin series is a way to express functions as infinite sums of their derivatives at zero. It is a special case of the Taylor series expanded around zero. This method is particularly useful when trying to approximate functions that are difficult to integrate exactly.
The Maclaurin series expansion for \(\sin(x)\) is given by \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\). When you square the series for \(\sin(x)\), you get the series for \(\sin^2(x)\).
To solve our problem, it's important to:

• Calculate the first few terms of the expansion because this usually provides a reasonable approximation for the function.
• Remember each term of the series is a polynomial, making integration straightforward.
• Check how the approximation converges to the exact function over the interval.

Using the first three terms allows us to compare the approximation's accuracy against the exact integral value. This comparison can illustrate how effective the Maclaurin series can be in practical applications.

Approximate Integration

Approximate integration is a method used when exact integration is complicated or impossible. Techniques like using partial sums from Maclaurin or Taylor series expansion can be very powerful.
When we integrate the first three terms of the Maclaurin series for \(\sin^2(x)\), we are effectively creating a polynomial approximation of the original function over a specific interval. This provides a means to approximate the area under the curve with a level of precision that can increase with more terms.

• The purpose is to simplify the integral into a form that is easier to compute.
• It provides insight into the nature of the function over a small range without solving a complex equation.
• Such approximation is very practical in numerical analysis and areas where computational resources are limited.

Understanding when and how to use approximate integration can be a tremendous asset in both theoretical and applied mathematics, particularly when you need to balance precision with computational efficiency.