Problem 32 - Calculus Early Transcendentals

Question

Complete the following steps for the given function, interval, and value of \(n\) a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$f(x)=x^{2} \text { on }[0,4] ; n=4$$.

Step-by-Step Solution

Answer

Answer: The midpoint Riemann sum is 21.

Step 1: Sketch the graph of the function on the given interval

To sketch the graph of \(f(x) = x^2\) within the interval \([0,4]\), we plot points along the function within the range of interest. Since \(f\) is an even function, it is symmetric with respect to the y-axis, and as x increases, the value of the function increases as well.

Step 2: Calculate \(\Delta x\) and grid points \(x_{0}, x_{1}, \ldots, x_{n}\)

We have \(a = 0\), \(b = 4\), and \(n = 4\). To find \(\Delta x\), we use the formula: \(\Delta x = \frac{b-a}{n} = \frac{4-0}{4} = 1\). Now, we calculate the grid points by iterating from \(x_0 = a = 0\) to \(x_n = b\) with a step of \(\Delta x = 1\). Thus, the grid points are \(x_0 = 0\), \(x_1 = 1\), \(x_2 = 2\), \(x_3 = 3\), and \(x_4 = 4\).

Step 3: Illustrate the midpoint Riemann sum by sketching appropriate rectangles

To illustrate the midpoint Riemann sum, we determine the midpoint of each subinterval and evaluate the function at this point. The height of the rectangle within each subinterval is then determined by the value of the function at the midpoint of the subinterval. In our case the midpoints are the following: $$x_0^* = 0.5, \ x_1^* = 1.5, \ x_2^* = 2.5, \ x_3^* = 3.5$$. Now, we evaluate the function at each midpoint: $$f(x_0^*) = 0.25, \ f(x_1^*) = 2.25, \ f(x_2^*) = 6.25, \ f(x_3^*) = 12.25.$$ We sketch the rectangles with the heights given by these values within each subinterval.

Step 4: Calculate the midpoint Riemann sum

To calculate the midpoint Riemann sum, we multiply the height of each rectangle (the function evaluated at the midpoint of the subinterval) by its width (\(\Delta x\)) and sum these products: $$M_n = f(x^*_0) \Delta x + f(x^*_1) \Delta x + f(x^*_2) \Delta x + f(x^*_3) \Delta x = (0.25 + 2.25 + 6.25 + 12.25) \Delta x = (0.25 + 2.25 + 6.25 + 12.25)(1) = 21.$$ Thus, the midpoint Riemann sum is 21.

Key Concepts

Midpoint Method

The midpoint method is a technique for estimating the area under a curve, often used in calculus when evaluating definite integrals. It is a variation of the Riemann sum and focuses specifically on using the midpoint of each subinterval to calculate the height of rectangles, which approximate the area.

To apply the midpoint method, you follow these steps:

• Divide the interval into smaller equally-sized subintervals.
• Find the midpoint of each subinterval. This is calculated as the average of the endpoints of the subinterval.
• Evaluate the function at each midpoint to determine the height of the rectangle over that subinterval.
• Multiply this height by the width of the subinterval (denoted as \( \Delta x \)), then sum the areas of all rectangles.

This approach generally provides a better approximation of the area compared to using the left or right endpoints because it accounts for the behavior of the function over each subinterval rather than just one side.

In our exercise, for the function \( f(x) = x^2 \) over the interval \([0, 4]\), we used midpoints \(0.5, 1.5, 2.5,\) and \(3.5\) to find the heights of the rectangles, which resulted in a more accurate approximation.

Grid Points

Grid points are crucial when utilizing numerical methods like the Riemann sum. They are points at which calculations are made across an interval, evenly dividing it into parts.

When you determine grid points:

• Identify the initial point \(a\) and the endpoint \(b\) of the interval.
• Decide on the number of subintervals \(n\) which dictates how the interval is split.
• Calculate \( \Delta x = \frac{b-a}{n} \) to find the width of each subinterval.
• The grid points \(x_0, x_1, x_2, ... , x_n\) follow from starting at \(a\) and adding \( \Delta x\) successively.

In this context, given \(a = 0\), \(b = 4\), and \(n = 4\), we find that \( \Delta x = 1\), leading to grid points at \(0, 1, 2, 3,\) and \(4\). These points help in structuring the entire interval to make useful calculations, enabling techniques like the midpoint method to be applied effectively.

Interval Calculation

Interval Calculation is a fundamental aspect of approximation techniques like the Riemann sum. It involves partitioning an interval into smaller, more manageable pieces, each of which is of equal width, denoted as \( \Delta x \).

Here's how to calculate intervals:

• Determine the total length of the interval by subtracting the lower bound \(a\) from the upper bound \(b\).
• Decide the number \(n\) of subintervals desired for your calculation. More subintervals often result in a more accurate approximation but require more computations.
• Calculate \( \Delta x \) as \( \Delta x = \frac{b-a}{n} \).

With \(f(x) = x^2\) on \([0, 4]\) and \(n = 4\), the interval calculation yields \( \Delta x = 1\). This regular partitioning ensures that each subinterval is equal in size, which simplifies the computation of numerical approximations like the midpoint Riemann sum, providing a clear and structured approach to solving integrals approximately.