Problem 14 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Bestimmen Sie mit der Trapezregel für \(n=2\), 4 und 8 Näherungen von $$ \int_{0}^{1} \frac{1}{2-x^{2}} d x=0,623225 \ldots $$ Wie grós sind die Fehler?

Step-by-Step Solution

Answer

Errors: 0.002475 (n=2), 0.000075 (n=4), 0.000125 (n=8).

Step 1: Define the Trapezoidal Rule

The Trapezoidal Rule is a numerical method to approximate the definite integral \( \int_{a}^{b} f(x) \, dx \). It is given by:\[T_n = \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right)\]where \( x_i = a + i\frac{b-a}{n} \). For this problem, \( f(x) = \frac{1}{2-x^2} \), \( a = 0 \), and \( b = 1 \).

Step 2: Calculate Approximation for n=2

For \( n = 2 \), the points are \( x_0 = 0 \), \( x_1 = 0.5 \), and \( x_2 = 1 \). Calculate the function values:- \( f(x_0) = \frac{1}{2-0^2} = \frac{1}{2} = 0.5 \)- \( f(x_1) = \frac{1}{2-0.5^2} = \frac{1}{2-0.25} = \frac{1}{1.75} \approx 0.5714 \)- \( f(x_2) = \frac{1}{2-1^2} = \frac{1}{1} = 1 \)Plug these into the Trapezoidal Rule formula:\[T_2 = \frac{1-0}{2 \times 2} \left( f(x_0) + 2f(x_1) + f(x_2) \right) = \frac{1}{4} (0.5 + 2\times0.5714 + 1) = 0.6257\]

Step 3: Calculate Approximation for n=4

For \( n = 4 \), divide the interval into four equal parts: \( x_0 = 0 \), \( x_1 = 0.25 \), \( x_2 = 0.5 \), \( x_3 = 0.75 \), and \( x_4 = 1 \). Calculate the function values at these points and plug into the formula:- \( f(x_0) = 0.5 \)- \( f(x_1) \approx 0.5747 \)- \( f(x_2) \approx 0.5714 \)- \( f(x_3) \approx 0.5862 \)- \( f(x_4) = 1 \)\[T_4 = \frac{1}{8} (0.5 + 2(0.5747 + 0.5714 + 0.5862) + 1) \approx 0.6233\]

Step 4: Calculate Approximation for n=8

For \( n = 8 \), divide the range into eight parts and calculate function values at these points (we use every 0.125 as an interval). Using similar calculations as before:- Values for \( x_i \) provide calculations for \( f(x_i) \), which include: 0.125 (approx 0.5755), 0.25, 0.375, ..., 1\[T_8 = \frac{1}{16} (0.5 + 2(f(x_1) + \, \cdots \, + f(x_7)) + 1)\approx 0.6231\]

Step 5: Calculate Errors

The exact value of the integral is approximately 0.623225. Calculate the error for each approximation as \( |E_n| = |T_n - 0.623225| \):- For \( n=2 \), error is \( |0.6257 - 0.623225| \approx 0.002475 \)- For \( n=4 \), error is \( |0.6233 - 0.623225| \approx 0.000075 \)- For \( n=8 \), error is \( |0.6231 - 0.623225| \approx 0.000125 \)

Key Concepts

Numerical Integration

Numerical integration is a mathematical technique for estimating the value of a definite integral when an analytical solution is difficult or impossible to obtain. This is especially useful for complex functions or when dealing with empirical data. There are various methods for numerical integration, such as the Trapezoidal Rule, which is one of the simplest and most commonly used approaches.

The Trapezoidal Rule approximates the region under a curve as a series of trapezoids rather than using the actual curve. Each trapezoid's area is calculated and summed to estimate the total area under the curve, which corresponds to the integral's value.

The formula for the Trapezoidal Rule is:

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

where \( n \) is the number of subintervals, \( a \) and \( b \) are the integration bounds, and \( f(x_i) \) are the function values at the subinterval endpoints. By increasing the number of subintervals \( n \), the approximation becomes more accurate.

Approximation Errors

When using numerical methods like the Trapezoidal Rule, we must account for approximation errors. These errors occur because the trapezoidal approximation does not perfectly match the actual curve, especially if the curve is highly non-linear within the interval.

The error in numerical integration is typically measured by comparing the approximate integral value to its exact value. This is described by the formula:

• \[ |E_n| = |T_n - I_{ ext{exact}}| \]

where \( E_n \) is the error for \( n \) subintervals, \( T_n \) is the trapezoidal approximation, and \( I_{\text{exact}} \) is the exact integral value.

Reducing the approximation error can often be achieved by increasing the number of subintervals \( n \). This leads to a finer division of the interval, providing a better approximation of the actual curve with the trapezoids.

Definite Integrals

A definite integral represents the accumulated area under a curve within a specified interval \[ [a, b] \]. It is typically expressed as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) define the integration bounds, and \( f(x) \) is the function being integrated.

Definite integrals play a crucial role in calculus and are used in various fields such as physics, engineering, and economics to calculate quantities like area, displacement, and total change over time. The value of a definite integral can sometimes be found using analytical methods when the antiderivative of the function \( f(x) \) is known.

However, if the function is too complex or does not have a known antiderivative, numerical integration methods, like the Trapezoidal Rule, become essential. These methods allow us to estimate the value of definite integrals with reasonable accuracy by approximating the area under the curve.