Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $$ \int_{4}^{9} \sqrt{x} d x, n=8 $$

Step-by-Step Solution

Answer

The definite integral can be estimated with the trapezoidal and Simpson's rules. These approximations now need to be compared to the exact value obtained by directly integrating \(\sqrt{x}\). The exact results will depend on the specific calculation. The error in the approximation can be found by subtracting the approximated value from the exact value of the integral.

Step 1: Set up the problem

The interval of integration is from 4 to 9 and the number of subdivisions, \(n\), is 8. This means we'll need to divide the interval into 8 equal parts. That means the width, \(h\), of each interval segment is \((9 - 4) / 8 = 0.625\). The points at which we will evaluate the function will be \(x = 4, 4.625, 5.25, \ldots, 9\).

Step 2: Apply Trapezoidal Rule

The Trapezoidal Rule can be stated as \(\int_a^b \! f(x) \, dx \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2)+\ldots+2f(x_{n-1}) + f(x_n)]\). Applying this to the function \(\sqrt{x}\) from 4 to 9 with \(n=8\) gives the approximate integral value.

Step 3: Apply Simpson's Rule

Simpson's Rule can be stated as \(\int_a^b \! f(x) dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3)+\ldots+4f(x_{n-1}) + f(x_n)]\). When we substitute the values for this problem into this formula, we get an approximation of the definite integral by Simpson's rule.

Step 4: Calculate Exact Value

The exact value of a definite integral can be found by anti-differentiation. The antiderivative of \(\sqrt{x}\) is \(2/3 \cdot x^{3/2}\). We can find the exact value by substituting the limits 9 and 4 into the antiderivative function and subtracting.

Step 5: Compare Approximations with Exact Value

Both the Trapezoid and Simpson's rule provide us with an approximation of the integral. Now, their results must be compared with the exact value calculated in Step 4. The difference between the approximated and exact values gives a measure of the error in the approximation.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a technique for approximating the definite integral of a function. When the antiderivative is complex or unknown, this rule helps approximate the integral by dividing the area under the curve into trapezoids rather than perfect rectangles.
This approach is simple and involves:

• Dividing the interval from the lower bound to upper bound into equal smaller intervals.
• Calculating the height of each trapezoid, which is the value of the function at certain points within the intervals.
• Summing these trapezoid areas to approximate the total area under the curve.

For this exercise, with the function \sqrt{x} over the range from 4 to 9 divided into 8 equal parts, we use the Trapezoidal Rule formula: \[\int_a^b f(x) \, dx \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + f(x_n)],\] where \( h \) is the width of each interval, calculated here as 0.625.

Simpson's Rule

Simpson's Rule is another numerical integration technique used for estimating the definite integral of a function. Unlike the Trapezoidal Rule, which uses straight line segments (trapezoids) to approximate the area under a curve, Simpson's Rule utilizes parabolic arcs, offering a more accurate estimation when the function is smooth.
This method requires:

• Dividing the interval into an even number of subintervals (in this case, 8) to ensure accuracy.
• Applying the Simpson's Rule formula that weights the midpoint of each subinterval differently, following the pattern of \[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + f(x_n). \]

This pattern results in an approximation formula: \[\int_a^b f(x) \, dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + ... + f(x_n)].\] In this problem, the integral from 4 to 9 of \( \sqrt{x} \), partitioned into 8 subintervals, shows Simpson's Rule's enhanced accuracy compared to the Trapezoidal Rule.

Numerical Integration

Numerical Integration refers to techniques used for approximating the value of definite integrals when finding a closed form antiderivative is challenging or impossible. The methods serve as valuable tools in calculus, allowing us to handle complex and irregularly behaved functions.
Two popular methods include:

• The Trapezoidal Rule, which approximates the integral using trapezoidal areas.
• Simpson's Rule, which leverages quadratic parabolas for a more refined estimation.

Numerical methods prove crucial in real-world applications, providing estimates for measurements, engineering analyses, and simulations where exact values are unnecessary or unattainable.

Antiderivative

The antiderivative, also known as an indefinite integral, is a function whose derivative gives the original function. It plays a significant role in finding an exact value of a definite integral. Once the antiderivative is known, it allows for:

• Substituting the upper and lower bounds of the integration range.
• Calculating the exact value by evaluating the antiderivative at these points and subtracting the results.

In the exercise, the antiderivative of \( \sqrt{x} \) is \( \frac{2}{3} x^{3/2} \). Substituting the limits 9 and 4, you determine the exact value of the integral. This process illustrates how antiderivatives facilitate precise calculations in integral calculus, contrasting with the estimation provided by numerical methods.