Question

Riemann sums for larger values of \(n\) Complete the following steps for the given function \(f\) and interval. a. For the given value of \(n\), use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval. $$f(x)=x^{2}-1 \text { on }[2,5] ; n=75$$

Short answer

Question: Calculate the left, right, and midpoint Riemann sums for the function \(f(x) = x^2 - 1\) on the interval \([2,5]\) using \(n = 75\) subintervals. Estimate the area under the curve by averaging these three sums. Answer: To find the left, right, and midpoint Riemann sums, follow these steps: 1. Calculate the width of each subinterval: \(\Delta x = \frac{5-2}{75}\) 2. Use the formula for left Riemann sum: \(L_n = \frac{3}{75} \sum_{i=0}^{74} ((2+ \frac{3i}{75})^2 -1)\) 3. Use the formula for right Riemann sum: \(R_n = \frac{3}{75} \sum_{i=1}^{75} ((2+ \frac{3i}{75})^2 -1)\) 4. Use the formula for midpoint Riemann sum: \(M_n = \frac{3}{75} \sum_{i=0}^{74} ((2+ \frac{3}{150} + \frac{3i}{75})^2 -1)\) 5. Calculate the average of the three sums to estimate the area: \(A \approx \frac{L_n + R_n + M_n}{3}\) Compute the Riemann sums and average using a calculator to estimate the area under the curve.

Step-by-step solution

Setup subintervals and width

To set up the subintervals, first, we need to find the width of each subinterval. The width is given by the formula: $$\Delta x = \frac{b-a}{n}$$ Here, \(a=2\), \(b=5\), and \(n=75\). So, the width (\(\Delta x\)) is: $$\Delta x = \frac{5-2}{75}$$

Calculate left Riemann sum

To compute the left Riemann sum, we will use the left endpoint of each subinterval. The formula is: $$L_n = \Delta x \sum_{i=0}^{n-1} f(a+i\Delta x)$$ Using the given function, width, and interval, we get: $$L_n = \frac{3}{75} \sum_{i=0}^{74} ((2+ \frac{3i}{75})^2 -1)$$ Now, compute this sum using a calculator.

Calculate right Riemann sum

For the right Riemann sum, we will use the right endpoint of each subinterval. The formula is: $$R_n = \Delta x \sum_{i=1}^{n} f(a+i\Delta x)$$ Using the given function, width, and interval, we get: $$R_n = \frac{3}{75} \sum_{i=1}^{75} ((2+ \frac{3i}{75})^2 -1)$$ Now, compute this sum using a calculator.

Calculate midpoint Riemann sum

For the midpoint Riemann sum, we will use the midpoint of each subinterval. The formula is: $$M_n = \Delta x \sum_{i=0}^{n-1} f(a+ \frac{1}{2}\Delta x + i\Delta x)$$ Using the given function, width, and interval, we get: $$M_n = \frac{3}{75} \sum_{i=0}^{74} ((2+ \frac{3}{150} + \frac{3i}{75})^2 -1)$$ Now, compute this sum using a calculator.

Estimate the area

Based on the approximations found for the left, right, and midpoint Riemann sums, you can use their average to estimate the area of the region bounded by the graph of \(f\) and the \(x\)-axis on the interval. The formula is: $$A \approx \frac{L_n + R_n + M_n}{3}$$ After calculating the sums in Steps 2, 3, and 4, substitute their values into this formula and compute the estimated area.

Key concepts

Sigma Notation

Sigma notation, denoted by the Greek letter \( \Sigma \), is a concise way to represent the sum of a sequence of numbers. In the context of Riemann sums, sigma notation plays a crucial role in expressing the sum of function values at specific points of an interval. For example, the formula \( \sum_{i=1}^{n} a_i \) tells us to sum the sequence of numbers \( a_1, a_2, ..., a_n \). In the case of Riemann sums, the \( a_i \) would represent the value of the function at a certain point, multiplied by the width of the subintervals.

When working with Riemann sums, the sigma notation allows us to write complex summations neatly and provides a general form that can be computed for any number of partitions \( n \). This makes calculating areas under curves more manageable, especially as \( n \) becomes very large, which is common in these applications.

Left Riemann Sum

A Left Riemann Sum is an approximation of the area under a curve by summing the areas of rectangles that touch the curve at their left-hand corners. In each subinterval \( [x_{i-1}, x_i] \), we take the height of the rectangle to be the function value at \( x_{i-1} \), the left endpoint of the subinterval.

The general formula is \( L_n = \Delta x \sum_{i=0}^{n-1} f(a+i\Delta x) \) where \( \Delta x \) is the width of the subintervals, and \( n \) is the number of them. Without a calculator, setting up the left Riemann sum involves understanding how to multiply each function value by the subinterval width and then using sigma notation for summing these products from the first to the \( (n-1) \)th interval.

Right Riemann Sum

Conversely, a Right Riemann Sum uses the right-hand corners of the subintervals to approximate the area under a curve. Here, the height of each rectangle is determined by the function value at the right endpoint \( x_i \). The formula analogous to the left sum is: \( R_n = \Delta x \sum_{i=1}^{n} f(a+i\Delta x) \).

When calculating the right Riemann sum, attention must be given to the index of summation starting at 1 and ending at \( n \) since it includes the right endpoint of the last subinterval. This subtle difference can be significant when comparing the right and left sum approximations of an area under a curve.

Midpoint Riemann Sum

The Midpoint Riemann Sum improves upon the left and right sums by taking the average of the function's value at the midpoints of each subinterval. This often results in a more accurate approximation of the area under the curve. The formula for the midpoint sum is \( M_n = \Delta x \sum_{i=0}^{n-1} f(a+ \frac{1}{2}\Delta x + i\Delta x) \).

To utilize this method, one must calculate the midpoint between \( x_{i-1} \) and \( x_i \) for each subinterval and evaluate the function at this point. The versatility of the midpoint sum, which generally falls between the left and right sum estimations, can provide a balanced approach to area approximation.

Area Approximation Methods

Approximating the area under a curve is a fundamental application of calculus, and Riemann sums (left, right, and midpoint) are the foundational techniques. By dividing the region into subintervals and summing the areas of rectangles, trapezoids, or other shapes that fit under the curve, we can estimate the area with varying degrees of accuracy.

Riemann sums are particularly useful because they become exact as the number of subintervals increases indefinitely (lim \( n \rightarrow \infty \)). Additionally, they provide a geometric interpretation of the definite integral as \( n \) approaches infinity. Practically, we often settle for an approximation, which can be improved by increasing \( n \) or using more sophisticated methods like Simpson's rule or the trapezoidal rule for better precision.