Question

Approximate the integral to three decimal places using the indicated rule. \(\int_{0}^{1} \sin ^{2}(\pi x) d x ;\) trapezoidal rule; \(n=6\)

Short answer

The approximate integral is 0.500.

Step-by-step solution

Understand the Trapezoidal Rule

The trapezoidal rule is a numerical method to approximate the definite integral of a function. When using it, the interval \([a, b]\) is divided into \(n\) equal subintervals of width \(h = \frac{b-a}{n}\). Each subinterval forms a trapezoid, and the area of these trapezoids is used to approximate the integral.

Define the Parameters

For the integral \(\int_{0}^{1} \sin^2(\pi x) \, dx\), the parameters are \(a = 0\), \(b = 1\), and \(n = 6\). We first calculate the subinterval width \(h = \frac{1-0}{6} = \frac{1}{6}\).

Compute Subinterval Points

Calculate the points at which the function will be evaluated. These points are \(x_0, x_1, \ldots, x_n\), where \(x_i = a + i \cdot h\). Thus, we have \(x_0 = 0, x_1 = \frac{1}{6}, x_2 = \frac{2}{6}, x_3 = \frac{3}{6}, x_4 = \frac{4}{6}, x_5 = \frac{5}{6}, x_6 = 1\).

Evaluate the Function at Each Point

Calculate the value of the function \(\sin^2(\pi x)\) at each point \(x_i\). \[ \begin{align*} f(x_0) &= \sin^2(0 \cdot \pi) = 0, \ f(x_1) &= \sin^2\left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}, \ f(x_2) &= \sin^2\left(\frac{2\pi}{6}\right) = \sin^2\left(\frac{\pi}{3}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}, \ f(x_3) &= \sin^2\left(\frac{3\pi}{6}\right) = \sin^2\left(\frac{\pi}{2}\right) = 1, \ f(x_4) &= \sin^2\left(\frac{4\pi}{6}\right) = \sin^2\left(\frac{2\pi}{3}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}, \ f(x_5) &= \sin^2\left(\frac{5\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}, \ f(x_6) &= \sin^2(\pi) = 0. \end{align*} \]

Apply the Trapezoidal Rule Formula

The trapezoidal approximation is given by \[ T_n = \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]. Substitute the values: \[ T_6 = \frac{1/6}{2} \left( 0 + 2 \left( \frac{1}{4} + \frac{3}{4} + 1 + \frac{3}{4} + \frac{1}{4} \right) + 0 \right) = \frac{1}{12} \left( 0 + 2 \cdot 3 + 0 \right) = \frac{1}{12} \cdot 6 = \frac{1}{2} \equiv 0.5. \]

Round to Required Decimal Places

The final result, already a two-decimal number, is 0.5. Since three decimals are requested, express it as 0.500.

Key concepts

Numerical Integration

Numerical integration is a useful method when it comes to approximating the value of definite integrals. It is particularly handy when the antiderivative of a function is difficult or impossible to find analytically. When a definite integral needs to be evaluated, numerical techniques provide a reliable way to estimate the integral over a certain interval. These methods are based on approximating the area under a curve, which is essentially what an integral represents.

There are several methods of numerical integration, with the trapezoidal rule being one of the simplest and most straightforward approaches. This is because it approximates the region under the curve as a series of trapezoids rather than more complex geometric shapes. Understanding how numerical integration works in practical scenarios can greatly aid in solving various problems in physics, engineering, and other scientific fields where exact integration might be too complex.

Definite Integral

A definite integral is an integral that is evaluated over a specific interval \( [a, b] \). It represents the signed area under a curve of a function \( f(x) \) from \( x = a \) to \( x = b \). The operation of finding the definite integral involves calculating this area, which can sometimes be complex, hence the usefulness of numerical methods.

The process of evaluating a definite integral gives one a numerical value that is central in calculations in science and engineering—such as finding total displacement given a velocity function, or calculating work done by a variable force. In our example, we were tasked with approximating the integral \( \int_{0}^{1} \sin^{2}(\pi x) \, dx \), which signifies the area beneath the function \( \sin^{2}(\pi x) \) from 0 to 1. Understanding the definite integral as a concept helps in visualizing the idea of "area under the curve," which is a foundational notion in calculus.

Approximation Methods

Approximation methods are essential when tackling integrals that cannot be solved analytically. These methods offer a way to reach an answer that is close enough to the true value for practical purposes. The core idea is to use simpler mathematical operations to estimate more complex calculations. For integrals, approximation methods can involve different algorithms or rules, with the trapezoidal rule being among them.

The trapezoidal rule works by splitting the interval of integration into smaller subintervals and approximating the area under the curve as a series of trapezoids. It gives a result by summing up the areas of these trapezoids. By increasing the number of subintervals, we can usually increase the accuracy of our approximation, though this may also increase computational effort. The balance between accuracy and computational efficiency is a key aspect of approximation methods.

Subinterval Width

The subinterval width, denoted as \( h \), plays a crucial role in the trapezoidal rule and other numerical integration techniques. It determines how finely the interval \( [a, b] \) is divided. Mathematically, it is calculated as \( h = \frac{b-a}{n} \), where \( n \) is the number of subintervals.

The choice of \( n \) significantly affects the approximation accuracy. A smaller \( h \) typically yields a more accurate result since the area under the curve is represented by more trapezoids that better fit the function's shape. However, a smaller \( h \) means more calculations, which can be computationally demanding. In our example, the interval from 0 to 1 was divided into 6 equal subintervals, resulting in \( h = \frac{1}{6} \). This granularity helps in making a good approximation of the integral.