Problem 33 - Calculus

Question

Approximate \(\int_{1}^{2} \frac{1}{1+x^{4}} d x\) using the Trapezoidal Rule with \(n=8\), and give an upper bound for the absolute value of the error.

Step-by-Step Solution

Answer

Using the Trapezoidal Rule with \( n=8 \), the approximate integral is calculated by evaluating the sum with calculated function values and determining error bound using the second derivative.

Step 1: Identify the Function and Limits

The function we are integrating is \( f(x) = \frac{1}{1 + x^4} \). The limits of integration are from \( a = 1 \) to \( b = 2 \). We will use the Trapezoidal Rule to approximate the integral.

Step 2: Calculate Step Size

The number of subintervals \( n \) is 8. The step size \( h \) is calculated as \( h = \frac{b-a}{n} = \frac{2-1}{8} = \frac{1}{8} \).

Step 3: Determine the x-values

Using the step size \( h = \frac{1}{8} \), compute the x-values: \( x_0 = 1 \), \( x_1 = 1.125 \), \( x_2 = 1.25 \), \( x_3 = 1.375 \), \( x_4 = 1.5 \), \( x_5 = 1.625 \), \( x_6 = 1.75 \), \( x_7 = 1.875 \), \( x_8 = 2 \).

Step 4: Evaluate the Function at the x-values

Calculate the values of \( f(x) \) at each \( x \) point: \( f(x_0) = \frac{1}{1 + 1^4} \), \( f(x_1) = \frac{1}{1 + 1.125^4} \), \( \ldots \), \( f(x_8) = \frac{1}{1 + 2^4} \).

Step 5: Apply the Trapezoidal Rule Formula

The formula for the Trapezoidal Rule is given by \[ T_n = \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]. Substitute \( h \) and the calculated function values to obtain the approximation.

Step 6: Evaluate the Sum

Calculate \( T_8 = \frac{1}{16} [ f(x_0) + 2(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6)+f(x_7)) + f(x_8) ] \) using the computed \( f(x) \) values.

Step 7: Find the Second Derivative

Calculate \( f''(x) \) for error estimation. \( f'(x) = \frac{-4x^3}{(1+x^4)^2} \), and then \( f''(x) = \frac{12x^6 - 12x^2(1+x^4)}{(1+x^4)^3} \).

Step 8: Determine Maximum of Second Derivative

Evaluate \( f''(x) \) within the interval \([1, 2]\) to find its maximum for error bound calculation. This requires checking endpoints or critical points by solving \( f''(x) = 0 \).

Step 9: Calculate Error Bound

Use the error bound formula for the Trapezoidal Rule: \( |E_T| \leq \frac{(b-a)^3}{12n^2} \max |f''(x)| \) for \( x \in [a,b] \). Substitute the maximum found in the previous step to find the upper bound for the error.

Key Concepts

Numerical Integration

Numerical Integration is an essential concept in calculus that allows us to approximate the values of definite integrals. While we can solve some integrals analytically, others may not have a closed-form solution, making numerical methods a useful tool. The Trapezoidal Rule is one of the simplest numerical integration methods.

• The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids rather than rectangles.
• It uses line segments to connect the points along the curve, simplifying calculations compared to more complex methods.
• This rule is particularly effective when the function being integrated is reasonably smooth and continuous over the interval of interest.

To apply the Trapezoidal Rule, you determine the number of subintervals, calculate the function values at each subinterval point, and then use the trapezoidal formula to find your approximation. The formula for the Trapezoidal Rule is: \[ T_n = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]where, \(h\) is the width of each subinterval, and \(x_0, x_1, ..., x_n\) are the x-values at the subinterval endpoints.

Error Estimation

Error Estimation is crucial when using numerical methods to approximate integrals, as it helps us understand how close our numerical result is to the true value. The Trapezoidal Rule has its own formula for estimating this error.
For the Trapezoidal Rule, the error estimation formula is:\[ |E_T| \leq \frac{(b-a)^3}{12n^2} \max |f''(x)| \]

• Here, \(|E_T|\) represents the absolute value of the error in the approximation.
• \(b-a\) is the width of the entire integral interval.
• \(n\) is the number of subintervals.
• \(\max |f''(x)|\) denotes the maximum absolute value of the second derivative of the function over the interval \([a,b]\).

By calculating the second derivative and determining its maximum value over the interval, we obtain a careful estimation of the error. This process reassures us that the result is within a specific bound, providing confidence in the numerical approximation.

Calculus

Calculus is the branch of mathematics dealing with continuous change, and it forms the foundation for methods such as the Trapezoidal Rule in numerical integration.
When we discuss calculus, there are two main parts: differential calculus and integral calculus.

• Differential Calculus: This focuses on the concept of the derivative, which represents rates of change and slopes of curves. For example, when calculating the second derivative, as done in the error estimation step, we use differential calculus principles.
• Integral Calculus: Integral calculus involves the concept of accumulating quantities, such as areas under curves, which is essentially what the Trapezoidal Rule helps us calculate.

In the context of the problem, we see the application of both parts of calculus: computing derivatives for error estimation and approximating a definite integral numerically. This interaction shows the power and necessity of calculus in solving practical mathematical problems.