Problem 2 - Calculus

Question

In Problems \(1-6,\) use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{3}} d x $$

Step-by-Step Solution

Answer

The exact integral value is \( \frac{4}{9} \).

Step 1: Divide the interval and calculate \\ \Delta x

The integral ranges from 1 to 3, and we're asked to use \( n = 8 \) subintervals. Calculate \( \Delta x \) as follows: \[ \Delta x = \frac{3 - 1}{8} = \frac{1}{4} \] This means each subinterval is 0.25 units wide.

Step 2: Calculate the left Riemann sum

The left endpoints are \( x_0 = 1, x_1 = 1.25, \ldots, x_7 = 2.75 \). The function is \( f(x) = \frac{1}{x^3} \). The left Riemann sum is:\[ L_n = \sum_{i=0}^{7} f(x_i) \Delta x \]\[ = \Delta x \left( f(1) + f(1.25) + \cdots + f(2.75) \right) \]Evaluate this by substituting \( f(x) \) and \( \Delta x = \frac{1}{4} \).

Step 3: Calculate the right Riemann sum

The right endpoints are \( x_1 = 1.25, x_2 = 1.5, \ldots, x_8 = 3 \). The right Riemann sum is:\[ R_n = \sum_{i=1}^{8} f(x_i) \Delta x \]\[ = \Delta x \left( f(1.25) + f(1.5) + \cdots + f(3) \right) \]Plug in the values for each \( f(x) \) and use \( \Delta x = \frac{1}{4} \).

Step 4: Calculate the trapezoidal rule

The trapezoidal rule formula is:\[ T_n = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_7) + f(x_8) \right) \]Here, \( f \) is evaluated at both endpoints and all intermediate points are doubled. Solve using the function \( f(x) = \frac{1}{x^3} \) and \( \Delta x = \frac{1}{4} \).

Step 5: Calculate the parabolic (Simpson's) rule

Simpson’s Rule requires that \( n \) is even, which it is. The formula is:\[ S_n = \frac{\Delta x}{3} \left( f(x_0) + 4(f(x_1) + f(x_3) + \cdots + f(x_7)) + 2(f(x_2) + f(x_4) + \cdots + f(x_6)) + f(x_8) \right) \]Calculate this using \( f(x) = \frac{1}{x^3} \) and \( \Delta x = \frac{1}{4} \).

Step 6: Calculate the exact value using the Second Fundamental Theorem of Calculus

To find the exact value, evaluate the integral \( \int_{1}^{3} \frac{1}{x^{3}} dx \). The antiderivative of \( \frac{1}{x^3} \) is \( -\frac{1}{2x^2} \). Use the bounds:\[ \left[ -\frac{1}{2x^2} \right]_{1}^{3} = -\frac{1}{2(3)^2} + \frac{1}{2(1)^2} \]\[ = -\frac{1}{18} + \frac{1}{2} \]\[ = \frac{9}{18} - \frac{1}{18} \]\[ = \frac{8}{18} = \frac{4}{9} \]Thus, the exact value of the integral is \( \frac{4}{9} \).

Key Concepts

Riemann Sum

The Riemann Sum is a foundational method in calculus used to approximate the value of a definite integral. It works by dividing the area under a curve into small rectangles or other shapes, summing their areas to approximate the total. For this, the interval between the lower and upper bounds (here 1 to 3) is split into numerous subintervals, each of width \( \Delta x \). In this exercise, with \( n = 8 \), we calculated \( \Delta x = 0.25 \).There are two primary types of Riemann sums:

• Left Riemann Sum: Utilizes function values at the left endpoints of subintervals. It tends to underestimate if the function is increasing.
• Right Riemann Sum: Uses function values at the right endpoints, often overestimating for increasing functions.

For each Riemann sum, the function \( f(x) = \frac{1}{x^3} \) is evaluated at calculated points, then scaled by\( \Delta x \), and summed together.These methods are crucial for understanding how integrals mirror the cumulative net area under a curve based on discrete calculations.

Trapezoidal Rule

The Trapezoidal Rule offers another method for approximating definite integrals. This approach improves the Riemann Sum by considering the area as trapezoids rather than rectangles, thus offering a potentially better approximation. In this exercise, each slice of interval 1 to 3 becomes a trapezoid. The exact formula is \[ T_n = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_7) + f(x_8) \right) \].Here's how it works:

• The function \( f(x) = \frac{1}{x^3} \) is computed at each subinterval endpoint.
• Except for the first and last terms, all interior terms are doubled.

The beauty of the Trapezoidal Rule is that by capturing the linear change between point evaluators, it often produces a more accurate estimate than basic Riemann sums.This method connects the idea of approximating a shape with straight lines, adding flexibility to understanding integral approximations.

Simpson's Rule

Simpson's Rule offers an even more refined technique than the Trapezoidal Rule for integral approximation using parabolic arcs instead of straight lines. It works exceptionally well when the interval is divided into even subintervals, as in this exercise with \( n = 8 \). The formula is as follows:\[ S_n = \frac{\Delta x}{3} \left( f(x_0) + 4(f(x_1) + f(x_3) + \ldots + f(x_7)) + 2(f(x_2) + f(x_4) + \ldots + f(x_6)) + f(x_8) \right) \].Simpson's Rule divides the area under the curve into subsections of parabolic arches, providing a closer fit to curves than linear methods:

• Each subinterval contributes both a linear and quadratic component that arcs smoothly.
• It weights the intermediate function values by 4s and 2s alternately, which multiplies those values for different contributions to create a parabolic fit.

The rule is particularly impactful when dealing with smooth and continuous functions and gives an excellent estimate for periodic integrals, offering a wider lens into approximations where curvature is present.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a powerful tool that connects differentiation and integration, providing insight into calculating exact integral values. It states that if \( F(x) \) is an antiderivative of \( f(x) \),then the definite integral of\( f(x) \) from\( a \) to\( b \) equals \( F(b) - F(a) \).In our exercise, we found the antiderivative of\( f(x) = \frac{1}{x^3} \) to be \( F(x) = -\frac{1}{2x^2} \).Thus, evaluating the definite integral from 1 to 3, we derive:\[ \left[ -\frac{1}{2x^2} \right]_{1}^{3} = -\frac{1}{18} + \frac{1}{2} = \frac{4}{9} \].This theorem is invaluable for converting an intricate problem of calculating the net area under a curve to one of algebraic evaluation with antiderivatives. Its deep connection between calculus' dual processes—differentiation and integration—enriches our understanding and utility of mathematics in solving real-world problems.