Question

Find an upper bound for the error in estimating \(\int_{0}^{3}(5 x+4) d x\) using the trapezoidal rule with six steps.

Short answer

The upper bound for the error is 0, because the second derivative is zero throughout the interval.

Step-by-step solution

Understanding the Problem

We're asked to estimate the integral at hand using the trapezoidal rule. Also, we need to find an upper bound on the error in this estimate. This involves determining the error formula and calculating the necessary values.

Identify the Interval and Number of Steps

The interval of integration is [0, 3]. We divide this interval into 6 equal steps. The width of each step is \( h = \frac{b - a}{n} = \frac{3 - 0}{6} = 0.5 \).

Calculate the Second Derivative of the Function

The function to integrate is \( f(x) = 5x + 4 \). The first derivative is \( f'(x) = 5 \), and the second derivative, \( f''(x) \), is 0 since the function is linear.

Error Formula for the Trapezoidal Rule

The error formula for the trapezoidal rule is given by: \[ E_T = -\frac{(b-a)^3}{12n^2} f''(c) \]where \(c\) is some point in the interval [a, b]. Since \(f''(x) = 0\), the error \(E_T = 0\) regardless of \(c\).

Conclusion

Since the second derivative \( f''(x) \) is 0 in the entire interval, the upper bound for the error in estimating the integral using the trapezoidal rule with six steps is 0.

Key concepts

Error Estimation

Understanding error estimation is crucial when dealing with numerical methods like the trapezoidal rule. It helps us gauge the accuracy of our integral approximation. When we estimate integrals using numerical methods, there might be some difference between the estimated value and the actual integral value. This difference is called the "error."

For the trapezoidal rule, the error formula is:

• \[ E_T = -\frac{(b-a)^3}{12n^2} f''(c) \]

where \(E_T\) is the error, \(b\) and \(a\) are the limits of integration, \(n\) is the number of steps, and \(f''(c)\) is the second derivative at some point \(c\) within the interval.

The trapezoidal rule error depends heavily on the second derivative of the function. If the second derivative is zero, as with linear functions, the error is zero. Situation like this, as seen in the given exercise, occurs when estimating the integral \( \int_{0}^{3}(5 x+4) dx \). Since the function \(5x + 4\) is linear, the second derivative is zero leading to an error of zero. This shows how crucial the nature of the function is to error estimation.

Integration

Integration is the process of finding the total accumulation of quantities. It's essentially the opposite operation of differentiation. In mathematical terms, integration calculates the area under a curve. When dealing with simple functions, we can often find integrals using standard calculus techniques.

However, for more complex functions or data sets, analytical integration might be cumbersome or impossible. In such cases, numerical integration techniques like the trapezoidal rule become invaluable. These numerical methods provide an approximate value of the integral.

For instance, in the exercise \( \int_{0}^{3}(5x + 4) \, dx \), the analytical solution would simply involve finding a function whose derivative matches \(5x + 4\). However, by dividing this interval into smaller sub-intervals and applying the trapezoidal rule, we can achieve an approximate estimate. This is especially useful when dealing with tabulated data or complex functions without simple antiderivatives.

Numerical Methods

Numerical methods are techniques used to obtain approximate solutions to mathematical problems that might not be easily solvable analytically. These methods are essential in fields like engineering, physics, computer science, and economics, where exact solutions are either difficult or impossible to find.

The trapezoidal rule is a popular numerical method for integration. It estimates the area under a curve by approximating it with trapezoids. For each sub-interval of the entire integration range, if you plot the function, you'll see small trapezoids under the curve.
By using the formula:

• \[ \text{Area} = \frac{b-a}{n} \sum_{i=1}^{n} \left( \frac{f(x_{i-1}) + f(x_i)}{2} \right) \]

we can estimate the total integral. Here, we divide the interval into smaller segments, calculate the average of the function values at each segment's endpoint, and sum them up.

Numerical methods, including the trapezoidal rule, are invaluable for tackling real-world problems where precision up to a reasonable margin is more critical than an exact solution. They allow for flexible and practical problem-solving approaches, especially when dealing with complex or computationally expensive functions.