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.

Verified step by step guidance

1. 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.
2. 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].
3. 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).
4. 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.
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].

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve on a given interval by dividing the area into smaller subintervals. Each subinterval's area is estimated using the value of the function at specific points, such as the right endpoint, left endpoint, or midpoint. The sum of these areas provides an approximation of the integral of the function over the interval.

Subintervals

Subintervals are smaller segments into which a larger interval is divided to facilitate the approximation of areas or integrals. In this case, the interval [0, 2] is divided into four equal parts, each of width Δx = (2-0)/4 = 0.5. This division allows for a more manageable calculation of the area under the curve by evaluating the function at specific points within each subinterval.

Geometric Interpretation

The geometric interpretation of a Riemann sum involves visualizing the area under the curve as a series of rectangles. For a right Riemann sum, the height of each rectangle is determined by the function's value at the right endpoint of each subinterval. This visual representation helps in understanding how the approximation converges to the actual area as the number of subintervals increases.