{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.


ƒ(𝓍) = 3 √x on [0,4] ; n = 40
(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.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with estimating the area under the curve of the function ƒ(𝓍) = 3√x on the interval [0,4] using Riemann sums with n = 40 subintervals. This involves dividing the interval into smaller subintervals and summing the areas of rectangles under the curve.
2. Step 2: Divide the interval [0,4] into n = 40 subintervals. The width of each subinterval, Δx, is calculated as Δx = (b - a) / n, where a = 0 and b = 4. Substitute these values into the formula to find Δx.
3. Step 3: Determine the x-values for the endpoints of each subinterval. These x-values are given by x₀, x₁, x₂, ..., xₙ, where x₀ = a and xₙ = b. Use the formula xᵢ = a + iΔx (for i = 0, 1, 2, ..., n) to calculate the x-values.
4. Step 4: Evaluate the function ƒ(𝓍) = 3√x at the appropriate x-values. Depending on whether you are using left Riemann sums, right Riemann sums, or midpoint Riemann sums, evaluate ƒ(𝓍) at the left endpoints, right endpoints, or midpoints of each subinterval.
5. Step 5: Compute the Riemann sum. Multiply the function values by the width of the subintervals (Δx) and sum them up. The formula for the Riemann sum is Σ[ƒ(xᵢ)Δx], where the summation runs over all subintervals. This sum provides an approximation of the area under the curve.

Key Concepts

Riemann Sums

Riemann sums are a method for approximating the area under a curve by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function value at that point is multiplied by the width of the subinterval. The sum of these products gives an estimate of the total area. As the number of subintervals (n) increases, the approximation becomes more accurate.

Definite Integrals

Definite integrals represent the exact area under a curve between two points on the x-axis. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral of a function over an interval provides a precise value for the area, contrasting with Riemann sums, which provide an approximation that improves with larger n.

Limit of Riemann Sums

The limit of Riemann sums as the number of subintervals approaches infinity leads to the exact value of the definite integral. This concept is crucial in calculus, as it formalizes the transition from discrete approximations to continuous areas. Understanding this limit helps in grasping how integration works and why it is a fundamental tool for calculating areas under curves.