Question

Use the trapezoidal rule with four subdivisions to estimate \(\int_{0}^{0.8} x^{3} d x\). Compare this value with the exact value and find the error estimate.

Short answer

Approximate integral: 0.0832, Exact: 0.1024, Error: 0.0192.

Step-by-step solution

Set Up the Trapezoidal Rule

The trapezoidal rule provides an approximation for the definite integral of a function. If we need to approximate \( \int_{a}^{b} f(x) \, dx \) with \( n \) subdivisions, the formula is \[T_n = \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{k=1}^{n-1} f(x_k) + f(x_n) \right]\]. In this exercise, \( a = 0 \), \( b = 0.8 \), \( n = 4 \), and \( f(x) = x^3 \). First, compute the width of each subinterval, \( \Delta x = \frac{b-a}{n} = 0.2 \).

Calculate the Function Values

Identify the x-values at each endpoint of the subintervals. These points are:\[x_0 = 0, \quad x_1 = 0.2, \quad x_2 = 0.4, \quad x_3 = 0.6, \quad x_4 = 0.8\]Calculate the function values:\[f(x_0) = (0)^3 = 0, \quad f(x_1) = (0.2)^3 = 0.008, \quad f(x_2) = (0.4)^3 = 0.064\]\[f(x_3) = (0.6)^3 = 0.216, \quad f(x_4) = (0.8)^3 = 0.512\]

Apply the Trapezoidal Rule

Substitute the values into the trapezoidal formula:\[T_4 = \frac{0.8-0}{2 \times 4} \times \left[ f(0) + 2(f(0.2) + f(0.4) + f(0.6)) + f(0.8) \right]\]\[T_4 = \frac{0.8}{8} \times \left[ 0 + 2(0.008 + 0.064 + 0.216) + 0.512 \right]\]\[T_4 = 0.1 \times \left[ 0.008 + 0.064 + 0.216 \right] \times 2 + 0.512\]\[T_4 = 0.1 \times (1.088)\]\[T_4 = 0.0832\]

Calculate the Exact Integral

Calculate the exact value of the integral:\[\int_{0}^{0.8} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{0.8} = \left( \frac{(0.8)^4}{4} \right) - \left( \frac{(0)^4}{4} \right)\]\[= \frac{0.4096}{4} = 0.1024\]

Calculate the Error Estimate

The error estimate is the difference between the exact value and the trapezoidal estimate:\[\text{Error} = |\text{Exact} - T_4| = |0.1024 - 0.0832| = 0.0192\]

Key concepts

Numerical Integration

Numerical integration is a powerful method used to approximate the definite integral of a function, especially when finding the exact integral is difficult or impossible. One of the popular techniques for numerical integration is called the trapezoidal rule. This method uses trapezoids to estimate the area under a curve.

The concept is simple: imagine dividing the area under a curve into smaller sections (trapezoids), and by summing up their areas, you approximate the integral. This is perfect for functions that are hard to integrate analytically.

• Each trapezoid is formed between two consecutive points on the x-axis.
• The top of each trapezoid follows the curve of the function.
• Increasing the number of subdivisions (trapezoids), usually improves accuracy.

While exact calculations might involve complicated calculus techniques, the trapezoidal rule makes it feasible to approximate even with simple arithmetic.

Error Calculation

When using the trapezoidal rule or any numerical method, it's important to calculate the error, which is the difference between the estimated value and the actual exact value.

In the exercise, after applying the trapezoidal rule, we calculated the estimated integral as 0.0832. On evaluating the integral exactly, it was found to be 0.1024. The error is 0.0192, calculated simply as the absolute difference between these two values.

Error calculation helps assess the reliability of the numerical method:

• If the error is small, the estimate is more reliable.
• An unusually high error might indicate the need for more subdivisions or using a more sophisticated method.

Understanding and accounting for error ensures better accuracy in numerical integration.

Definite Integral Estimation

Definite integral estimation using numerical methods like the trapezoidal rule is a technique to calculate the area under a curve between two points.

Why use estimation? For many functions, finding an exact antiderivative might be complex or take extensive algebraic manipulation, making numerical methods like the trapezoidal rule very useful.

• The process starts by selecting the interval over which the integral needs to be estimated.
• Next, divide this interval into smaller, manageable subdivisions.
• The function values at these points are used to create trapezoids, each approximating a part of the total area.

The trapezoidal rule provided an estimate, which, compared to the actual calculation, was close enough to be useful. This makes the trapezoidal rule an essential tool in calculus for handling definite integrals, especially in real-world applications like physics and engineering, where exact calculations might not be feasible.