Question

Estimate the minimum number of subintervals needed to approximate the integral \(\int_{1}^{4}\left(5 x^{2}+8\right) d x\) with an error magnitude of less than \(0.0001\) using the trapezoidal rule.

Short answer

The minimum number of subintervals needed is 475.

Step-by-step solution

Understand the Trapezoidal Rule Formula

The Trapezoidal Rule is an approximation method for definite integrals, given by: \( T_n = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \), where \( \Delta x = \frac{b-a}{n} \). Here, \(a = 1\) and \(b = 4\). The error bound for the Trapezoidal Rule is given by: \( \text{Error} \leq \frac{(b-a)^3}{12n^2}M \), where \(M\) is the maximum value of \(|f''(x)|\) on the interval \([a, b]\).

Compute Second Derivative of the Function

Let's find the second derivative of the function \( f(x) = 5x^2 + 8 \). First derivative, \( f'(x) = 10x \).Second derivative, \( f''(x) = 10 \). Since \( f''(x) = 10 \) is constant, the maximum value of \(|f''(x)|\) on the interval \([1, 4]\) is \( M = 10 \).

Setup Error Inequality

The error inequality for this problem is given by:\[ \frac{(4-1)^3}{12n^2} \cdot 10 < 0.0001 \]Simplifying this results in:\[ \frac{27 \cdot 10}{12n^2} < 0.0001 \] which further simplifies to \[ \frac{270}{12n^2} < 0.0001 \].

Solve for Minimum n

We solve the inequality:\[ \frac{270}{12n^2} < 0.0001 \]Multiplying both sides by \( \12n^2\) gives:\[ 270 < 0.0012n^2 \]Dividing both sides by \( 0.0012 \) results in:\[ n^2 > 225000 \]Taking the square root of both sides:\[ n > \sqrt{225000} \approx 474.34 \]Thus, the minimum integer value for \( n \) is \( n = 475 \).

Key concepts

Error Estimation

When approximating the value of a definite integral using numerical methods like the trapezoidal rule, understanding the associated error is crucial. Error estimation helps in determining the precision of our approximation.In the context of the trapezoidal rule, the error can be estimated using a formula that involves the second derivative of the function being integrated. The error bound formula is: \( \text{Error} \leq \frac{(b-a)^3}{12n^2}M \), where \( M \) represents the maximum value of \(|f''(x)|\) on the interval \([a, b]\). This tells us how error decreases as the number of subintervals \( n \) increases.

• The smaller the error threshold, the more subintervals will be necessary for an accurate approximation.
• The error estimation helps decide the least number of subintervals needed to achieve a desired level of accuracy.

For our problem, we wanted the error less than \( 0.0001 \). By calculating \( M \) and rearranging to solve for \( n \), we ensure that our approximation meets the accuracy required. Always remember, though, that the computed error is an "upper bound," meaning the actual error of the trapezoidal approximation could be lower but not higher than this computed value.

Definite Integral Approximation

Calculating definite integrals exactly can sometimes be a daunting task. In cases where finding an exact integral is challenging or impossible, numerical methods come to the rescue.The trapezoidal rule is one such method used to approximate definite integrals. It works by dividing the region under the curve into trapezoids rather than rectangles. This approach can lead to a more accurate approximation because it better accommodates the curve's slope.For example, the integral \( \int_{1}^{4}(5x^2 + 8) dx \) was approximated using the trapezoidal rule by first determining the intervals \( x_0, x_1, \ldots, x_n \) and calculating function values at these points. By applying the trapezoidal formula, we summed the areas of each trapezoid created by these subintervals.

• The trapezoidal rule is especially useful when dealing with functions that can easily be evaluated at various points.
• This method improves in accuracy as the number of subintervals increases, reducing approximation error.

Being aware of both the function's behavior and the desired accuracy helps when setting up the intervals for computation.

Numerical Analysis

Numerical analysis is a cornerstone in the field of applied mathematics and computation. It involves designing algorithms to obtain approximate solutions to complex mathematical problems that are difficult or impossible to solve analytically. One area of numerical analysis is the numerical integration, where methods like the trapezoidal rule provide an efficient means to estimate definite integrals. Such methods are crucial in practical applications where analytic solutions are unavailable.

Importance in Numerical Integration

Numerical analysis for integration:

• Allows computation of integrals that model real world phenomena, such as physics and engineering problems.
• Offers a way to handle complex or irregular functions, which cannot be integrated directly using traditional calculus techniques.

For the trapezoidal rule, it provides a clearer geometric interpretation by using trapezoids, making it easier to visualize how the approximation is formed. Through numerical analysis, concepts that seem daunting become manageable. It allows scientists and engineers to simulate and understand systems and phenomena when experimental or exact solutions are impractical or infeasible.