Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


ƒ(𝓍) = 1/x on [1,6] ; n = 5


(d) Calculate the midpoint Riemann sum.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to calculate the midpoint Riemann sum for the function ƒ(𝓍) = 1/x over the interval [1,6] with n = 5 subintervals. A midpoint Riemann sum approximates the area under the curve by summing the areas of rectangles whose heights are determined by the function value at the midpoint of each subinterval.
2. Step 2: Divide the interval [1,6] into n = 5 equal subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (6 - 1)/5 = 1. This means each subinterval has a width of 1.
3. Step 3: Determine the midpoints of each subinterval. The subintervals are [1,2], [2,3], [3,4], [4,5], and [5,6]. The midpoints are calculated as the average of the endpoints of each subinterval: (1+2)/2 = 1.5, (2+3)/2 = 2.5, (3+4)/2 = 3.5, (4+5)/2 = 4.5, and (5+6)/2 = 5.5.
4. Step 4: Evaluate the function ƒ(𝓍) = 1/x at each midpoint. Substitute each midpoint value into the function to find the heights of the rectangles: ƒ(1.5), ƒ(2.5), ƒ(3.5), ƒ(4.5), and ƒ(5.5).
5. Step 5: Multiply each function value by the width of the subinterval, Δ𝓍 = 1, and sum the results to calculate the midpoint Riemann sum. The formula is: Midpoint Riemann Sum = Δ𝓍 * [ƒ(1.5) + ƒ(2.5) + ƒ(3.5) + ƒ(4.5) + ƒ(5.5)].

Key Concepts

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over a specified interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and then summing the products of these values and the widths of the subintervals. The midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function.

Midpoint Rule

The midpoint rule is a specific type of Riemann sum that uses the midpoint of each subinterval to estimate the area under a curve. For a function f(x) over an interval [a, b] divided into n equal parts, the midpoint of each subinterval is calculated, and the function is evaluated at these midpoints. The sum of these values, multiplied by the width of the subintervals, provides an approximation of the integral.

Definite Integral

The definite integral of a function over an interval [a, b] represents the net area under the curve of the function between the two points a and b. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫_a^b f(x) dx and can be computed using various techniques, including Riemann sums.