Question

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.
(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

Accepted Answer

Step 1: Understand the problem. We are tasked with estimating the area under the curve ƒ(𝓍) = x² + 2 on the interval [0, 2] using a right Riemann sum with n = 4 subintervals. A Riemann sum approximates the area under a curve by summing the areas of rectangles formed over subintervals. Step 2: Divide the interval [0, 2] into n = 4 subintervals. The width of each subinterval, Δx, is calculated as Δx = (2 - 0) / 4 = 0.5. The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2]. Step 3: For a right Riemann sum, the height of each rectangle is determined by the function value at the right endpoint of each subinterval. The right endpoints of the subintervals are x = 0.5, x = 1, x = 1.5, and x = 2. Evaluate ƒ(𝓍) = x² + 2 at these points: ƒ(0.5), ƒ(1), ƒ(1.5), and ƒ(2). Step 4: Compute the area of each rectangle. The area of a rectangle is given by height × width. For each subinterval, the width is Δx = 0.5, and the height is the function value at the right endpoint. The areas are: A₁ = ƒ(0.5) × 0.5, A₂ = ƒ(1) × 0.5, A₃ = ƒ(1.5) × 0.5, and A₄ = ƒ(2) × 0.5. Step 5: Add the areas of all rectangles to approximate the total area under the curve. The right Riemann sum is given by: R₄ = A₁ + A₂ + A₃ + A₄. This sum provides the estimated area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2].

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.
(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

Risposta video verificata per un problema simile:

Concetti chiave

Riemann Sum

Subintervals

Geometric Interpretation

Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.