Question

Use the midpoint rule with eight subdivisions to estimate \(\int_{2}^{4} x^{2} d x\).

Short answer

The estimated value of the integral using the midpoint rule is approximately 18.65625.

Step-by-step solution

Determine the interval width

The integral is evaluated from 2 to 4. Calculate the width of each subinterval by using the formula \( \Delta x = \frac{b-a}{n} \), where \( a = 2 \), \( b = 4 \), and \( n = 8 \). Thus, \( \Delta x = \frac{4-2}{8} = \frac{1}{4} \).

Identify the midpoints of the subintervals

Since we are using the midpoint rule, determine the midpoints of the subintervals. The subintervals are [2, 2.25], [2.25, 2.5], ..., [3.75, 4]. The midpoints are 2.125, 2.375, 2.625, 2.875, 3.125, 3.375, 3.625, and 3.875.

Evaluate the function at midpoints

Evaluate the function \( f(x) = x^2 \) at each midpoint. This yields:- \( f(2.125) = (2.125)^2 = 4.515625 \)- \( f(2.375) = (2.375)^2 = 5.640625 \)- \( f(2.625) = (2.625)^2 = 6.890625 \)- \( f(2.875) = (2.875)^2 = 8.265625 \)- \( f(3.125) = (3.125)^2 = 9.765625 \)- \( f(3.375) = (3.375)^2 = 11.390625 \)- \( f(3.625) = (3.625)^2 = 13.140625 \)- \( f(3.875) = (3.875)^2 = 15.015625 \).

Apply the midpoint rule formula

The midpoint rule approximates the integral using \( M_n = \Delta x \times \sum_{i=1}^{n} f(m_i) \), where \( m_i \) are the midpoints. In this case, \( \Delta x = \frac{1}{4} \), thus:\[M_8 = \frac{1}{4} \times (4.515625 + 5.640625 + 6.890625 + 8.265625 + 9.765625 + 11.390625 + 13.140625 + 15.015625)\], which equals \[\frac{1}{4} \times 74.625 = 18.65625 \].

Key concepts

Midpoint Rule

The Midpoint Rule is a technique used in calculus for approximating the value of a definite integral. It is especially useful when an integral cannot be solved analytically or when only a numerical approximation is needed. The method approximates the area under a curve by dividing the interval into equal subdivisions and then using the midpoint of each subinterval to evaluate the function. This ensures that the height of the rectangle is representative of the function's value in the midpoint of that interval.
To apply the Midpoint Rule, follow these steps:

• Divide the interval \( [a, b] \) into n equal subintervals.
• Calculate the midpoint of each subinterval.
• Evaluate the function at each midpoint.
• Multiply the function values by the width of the subinterval and sum them all together.

This method is beneficial because it generally provides a more accurate approximation than the Left or Right Riemann Sum, particularly when the function is smooth over the interval.

Definite Integral

The Definite Integral is a fundamental concept in calculus, representing the accumulation of quantities, such as areas under curves. It is denoted by \( \int_{a}^{b} f(x) dx \)\ and signifies the total area between the x-axis and the function f(x) from x = a to x = b.
In the context of definite integration, a few key points should be highlighted:

• The integral's limits of integration (lower and upper limits) determine the interval over which the integration occurs.
• For positive functions, the definite integral represents the total area under the curve above the x-axis.
• It can also be interpreted geometrically as the signed area, taking into account areas above and below the x-axis resulting in positive and negative contributions.

In practical applications, the definite integral can model numerous phenomena, ranging from physical quantities like distance and area to more abstract calculations in probability and economics.

Numerical Integration

Numerical Integration is a powerful branch of numerical analysis focused on calculating the value of integrals, particularly when they are not easily solvable with analytical methods. This involves computing an approximation to the integral using discrete data points rather than a continuous function.
Here are some key methods within numerical integration:

• Trapezoidal Rule: Approximates the area under the curve using a series of trapezoids.
• Simpson's Rule: Utilizes parabolic arcs instead of straight lines to approximate the areas under curves.
• Midpoint Rule: Focuses on using midpoints of subintervals for a more balanced approximation.

The choice of method depends on factors like the desired accuracy and the function's integrability. Numerical integration is essential in many fields, from engineering to finance, where it helps solve real-world problems.

Subdivision

Subdivision in calculus refers to dividing an interval into smaller parts, or subintervals, which are then used to apply numerical integration techniques. It is a critical step in methods such as the Midpoint Rule, Trapezoidal Rule, or Simpson's Rule.
Key aspects of subdivision include:

• The number of subintervals \( n \) determines the accuracy; more subintervals generally lead to a better approximation of the integral.
• Each subinterval has a width calculated as \( \Delta x = \frac{b-a}{n} \), allowing the entire interval \( [a, b] \) to be covered fully.
• Subdivision helps to break down complex problems, making them manageable by focusing on smaller sections at a time.

Think of subdivision as slicing a cake into smaller pieces, where each piece is easier to analyze and sum up to get the whole. It's a simple yet indispensable concept in numerical methods, offering flexibility in computations.