Problem 1 Berechnen Sie \(I=\int_{0}^{\sqr... [FREE SOLUTION] (2024)

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Chapter 9: Problem 1

Berechnen Sie $I=\int_{0}^{\sqrt{3} / 3} \frac{\mathrm{d} x}{1+x^{2}}$ mitHilfe a) der Sehnentrapezformel; b) der Simpsonschen Formel. Führen Sie die Rechnung mit einem Taschenrechner durch und schätzen Sie denFehler ab. Vergleichen Sie die Näherungswerte mit dem exakten Wert desIntegrals.

Short Answer

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The main steps for solving this problem include calculating the exact integral value, applying the Trapezoidal and Simpson's Rule to approximate the integral value, and then calculating and understanding the error for these approximations. Finally, these values are compared to identify which approximation provides a result closer to the exact value.

Step by step solution

Calculate exact integral

First we need to find the exact integral of the given function. Use the antiderivative formula for $arctan(x)$ which is $\int \frac{1}{1+x^{2}}dx = arctan(x) +C$. Then calculate $I = arctan(\sqrt{3}/3) - arctan(0)$. This is the exact value of the integral for comparison in the following steps.

Apply Trapezoidal Rule

To apply the Trapezoidal Rule, we need to divide the given interval $[0,\sqrt{3} / 3]$ into equal subintervals. The general formula for the Trapezoidal Rule is $\frac{b-a}{2} [f(a) + f(b)]$. Apply this formula to the function $f(x) = 1/(1+x^{2})$ and the interval from 0 to $\sqrt{3}/3$. This gives the approximation of the integral under the trapezoidal rule.

Apply Simpson's Rule

For Simpson's rule, we also divide the interval into equal subintervals. The general formula is $\frac{b-a}{6} \[f(a) + 4f((a+b)/2) + f(b)\]$. We also apply this formula to the function $f(x) = 1/(1+x^{2})$ and the interval from 0 to $\sqrt{3}/3$. This gives the approximation of the integral under Simpson's rule.

Calculate error

Use the exact value from step 1 and subtract the values obtained in steps 2 and 3. This provides the approximated error for both the Trapezoidal Rule and Simpson's Rule.

Comparison

Finally, compare the values obtained from the Trapezoidal and Simpson's rules with the exact value. Also reflect on the error approximations. Discuss which approximation is closer and why that might be.

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$Problem 1 Berechnen Sie \(I=\int_{0}^{\sqr... [FREE SOLUTION] (3)$

Most popular questions from this chapter

Welche der folgenden uneigentlichen Integrale sind konvergent? a) $\int_{1}^{\infty} \frac{\mathrm{d} x}{x \cdot \mathrm{e}^{x}}$ b) $\int_{0}^{\infty} \mathrm{e}^{-x^{2}} \mathrm{~d} x$ c) $\int_{1}^{\infty} \frac{\sin x}{x^{2}} \mathrm{~d} x$ d) $\int_{1}^{\infty} \frac{x+1}{x^{2}} \mathrm{~d} x$Beweisen Sie: $\int_{0}^{\pi / 2} \sin ^{2 m} x \mathrm{~d} x=\frac{2 m-1}{2m} \cdot \frac{2 m-3}{2 m-2}, \ldots \cdot \frac{1}{2} \cdot \frac{\pi}{2}$für alle $m \in \mathbb{N}$.Berechnen Sie folgende Integrale mit Hilfe der summierten Sehnentrapezformel (6 Stellen nach dem Komma). Verwenden Sie 2,4 und 8 Teilintervalle und gebenSie zu a) eine Fehlerabschätzung an. a) $\int_{1}^{3} \ln x \mathrm{~d} x$ b) $\int_{1}^{1,5} \frac{\sin 2 \pi x}{x} \mathrm{~d} x$4\. Geben Sie eine untere und eine obere Schranke für $\int_{0}^{\pi / 2} \cosx \mathrm{~d} x$ an.

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