Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.PE.48

Chapter 8, Problem 8.PE.48

You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)

Verified step by step guidance

Recall the error bound formula for Simpson's Rule: the error \( E_S \) satisfies \( |E_S| \leq \frac{K(b - a)^5}{180 n^4} \), where \( K \) is an upper bound on \( |f^{(4)}(x)| \) over \([a, b]\), and \( n \) is the number of subintervals (which must be even).

Identify the given values: \( a = 1 \), \( b = 2 \), \( K = 3 \), and the desired error bound \( |E_S| < 10^{-5} \).

Set up the inequality for the error bound: \( \frac{3 (2 - 1)^5}{180 n^4} < 10^{-5} \).

Simplify the inequality to solve for \( n^4 \): \( \frac{3}{180 n^4} < 10^{-5} \) which leads to \( n^4 > \frac{3}{180 \times 10^{-5}} \).

Calculate the right side and then take the fourth root to find \( n \). Finally, choose the smallest even integer \( n \) that satisfies this inequality to ensure the error is less than \( 10^{-5} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simpson’s Rule

Simpson’s Rule is a numerical method for approximating definite integrals by fitting parabolas through subintervals of the function. It requires an even number of subintervals and generally provides more accurate results than the trapezoidal rule for smooth functions.

Error Bound for Simpson’s Rule

The error bound for Simpson’s Rule depends on the fourth derivative of the function. Specifically, the error magnitude is bounded by (K(b−a)^5) / (180 n^4), where K is the maximum absolute value of the fourth derivative on [a, b], and n is the number of subintervals.

Choosing Number of Subintervals for Desired Accuracy

To ensure the approximation error is below a specified tolerance, solve the error bound inequality for n, the number of subintervals. Since Simpson’s Rule requires n to be even, round up to the nearest even integer after calculating the minimum n needed.

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Textbook Question

4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.

Textbook Question

Heat capacity of a gas

Heat capacity

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.

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