Question

{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.

Accepted Answer

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. 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. 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. 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. 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.

{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.

Risposta video verificata per un problema simile:

Concetti chiave

Riemann Sums

Definite Integrals

Limit of Riemann Sums

{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.