Question

Use the trapezoidal rule to estimate \(\int_{0}^{1} x^{2} d x\) using four subintervals.

Short answer

The estimated integral using the trapezoidal rule is 0.34375.

Step-by-step solution

Determine the width of each subinterval

The width of each subinterval, denoted as \( h \), can be determined by the formula \( h = \frac{b-a}{n} \), where \( a \) and \( b \) are the limits of integration and \( n \) is the number of subintervals. Here, \( a = 0 \), \( b = 1 \), and \( n = 4 \). Thus, \( h = \frac{1-0}{4} = 0.25 \).

Identify the endpoints of each subinterval

The endpoints of each of the four subintervals are calculated by adding \( h \) successively starting from \( a = 0 \). The endpoints are: \( x_0 = 0 \), \( x_1 = 0.25 \), \( x_2 = 0.5 \), \( x_3 = 0.75 \), \( x_4 = 1 \).

Evaluate the function at each endpoint

Calculate \( f(x) = x^2 \) for each endpoint: \( f(x_0) = (0)^2 = 0 \), \( f(x_1) = (0.25)^2 = 0.0625 \), \( f(x_2) = (0.5)^2 = 0.25 \), \( f(x_3) = (0.75)^2 = 0.5625 \), and \( f(x_4) = (1)^2 = 1 \).

Apply the trapezoidal rule formula

The trapezoidal rule formula is: \[T = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right]\]Substituting the values, we get: \[T = \frac{0.25}{2} \left[ 0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1 \right]\]

Calculate the final estimate

Simplifying within the brackets, we have:\[ 0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1 = 0 + 0.125 + 0.5 + 1.125 + 1 = 2.75 \]Thus, \[ T = \frac{0.25}{2} \times 2.75 = 0.125 \times 2.75 = 0.34375 \].

Key concepts

Numerical Integration

In mathematics, numerical integration is a method used to estimate the value of a definite integral. This becomes especially useful when finding the exact integral value is difficult or impossible using standard analytical techniques. Numerical integration techniques approximate the area under a curve by breaking the area into smaller, manageable parts.

One of the primary techniques for numerical integration is the Trapezoidal Rule. The Trapezoidal Rule approximates the area under the curve as a series of trapezoids rather than rectangles, which often provides a better approximation than methods like the Riemann sum. Each trapezoid's area is calculated, then summed to estimate the total area under the curve. Because it uses trapezoids rather than straight lines, it can often capture the curve's shape better, leading to more accurate results when compared to other summation methods.

Subintervals

Subintervals are incredibly important in numerical integration. When using the Trapezoidal Rule, the interval over which you want to integrate is divided into smaller, equal-length segments, called subintervals. Each segment is defined by two neighboring points on the x-axis, often referred to as the endpoints of the subintervals.

In the problem at hand, the integral \(\int_{0}^{1} x^{2} \, dx \)is calculated using four subintervals. Each subinterval is \( h = \frac{1-0}{4} = 0.25 \) units wide. The division of the interval into four parts helps break down a complex problem, such as integration, into simpler, smaller problems that are easier to handle and solve. As you increase the number of subintervals, the approximations generally become closer to the actual value of the integral.

Definite Integrals

Definite integrals are a fundamental concept in calculus, representing the area under a curve bounded by two endpoints along the x-axis. It is described by an integral with limits, such as \(\int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration. Definite integrals can be interpreted geometrically as the net area between the graph of the function and the x-axis.

The definite integral gives us a precise way of accumulating quantities that have continuously varying rates, such as distance needs, which change depending on velocity over time. In practical problems like estimating the area under \(x^2\) between 0 and 1, the definite integral computes the exact area if done analytically. When it's not feasible to compute it exactly, however, methods such as the Trapezoidal Rule are employed for an approximate solution, as in this exercise.