Question

Find an upper bound for the error in estimating \(\int_{0}^{\pi} 2 x \cos (x) d x\) using Simpson's rule with four steps.

Short answer

Error bound: \(\frac{\pi^5}{720} \times \max_{0 \leq x \leq \pi} |16\cos(x)+8x\sin(x)|\).

Step-by-step solution

Define the Integral and the Function

The given integral is \(\int_{0}^{\pi} 2x \cos(x) \, dx\). Here, the integrand, which we will call \(f(x)\), is \(f(x) = 2x \cos(x)\). We'll use Simpson's Rule to approximate this integral.

State Simpson's Rule and Error Formula

Simpson's Rule is given by:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left( f(x_0) + 4 \sum_{\text{odd}} f(x_i) + 2 \sum_{\text{even}} f(x_i) + f(x_n) \right)\]where \(h = \frac{b-a}{n}\) and \(n\) is the number of intervals (which must be even). The error bound for Simpson's Rule is given by:\[E_s = \frac{(b-a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\]

Key concepts

Numerical Integration

Numerical integration is a powerful technique used to approximate the definite integral of a function when finding an exact solution analytically may be challenging, or when the function is specified only through discrete data points.
This technique transforms the continuous area under a curve into a sum of simpler geometric shapes like rectangles, trapezoids, or parabolas.
Among the various methods for numerical integration, Simpson's Rule is highly popular due to its accuracy and efficiency.

• Simpson's Rule approximates the function using parabolas over subintervals of the domain, rather than simple rectangles or trapezoids as in other methods.
• The rule requires the number of subintervals to be even and applies a formula that includes both the endpoints and the midpoints of these intervals to estimate the integral.
• This results in a more precise approximation compared to other simpler numerical integration methods like the Trapezoidal Rule.

Integrals are often used to determine areas and accumulation quantities, making accurate estimations crucial in fields ranging from physics to finance.

Error Estimation

Error estimation in numerical integration is crucial because it tells us how far off our approximation might be from the true value of the integral.
Simpson's Rule, while powerful, does not always provide the exact value, hence the need for estimating the possible error.
The error estimate formula for Simpson's Rule is essential because it helps assess the reliability of the approximation.

• The error bound for Simpson’s Rule, formulated as \(E_s = \frac{(b-a)^5}{180 n^4} \max_{a \leq x \leq b} |f^{(4)}(x)|\), allows us to calculate the maximum possible error associated with the numerical approximation.
• This formula incorporates the interval over which the integration is performed \((b-a)\), the number of subintervals \(n\), and importantly, the fourth derivative of the function \(f^{(4)}(x)\).
• The incorporation of the fourth derivative ensures that any rapid changes in the curvature of the function are accounted for, influencing the error estimate significantly.

By understanding and applying this error estimation, one can decide how many subintervals are needed to achieve a desired level of accuracy.

Fourth Derivative

The fourth derivative of a function, denoted as \(f^{(4)}(x)\), plays a crucial role in the error estimation of Simpson's Rule.
In calculus, higher-order derivatives provide detailed information about the rate of change and the curvature of a function.
For Simpson's Rule, particularly, the fourth derivative is used in the error term calculation.

• The significance of the fourth derivative in error estimation emerges from its ability to reflect the behavior of the function’s curvature.
• Curvature changes such as concavity and convexity are captured well by higher-order derivatives, helping predict the accuracy of the numerical approximation.
• The presence of the maximum value of \(f^{(4)}(x)\) in the error formula utilized in Simpson's Rule provides insight into the function’s potential rapid changes over the interval, which might lead to larger errors.

By calculating \(f^{(4)}(x)\), one can effectively gauge the behavior of the approximation error and thus control the precision of the integral estimation.