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.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.

Verified step by step guidance

1. First, divide the interval [0, 2] into 4 equal subintervals. Since the length of the interval is 2, each subinterval will have width \(\Delta x = \frac{2 - 0}{4} = 0.5\).
2. Identify the left endpoints of each subinterval. These will be \(x_0 = 0\), \(x_1 = 0.5\), \(x_2 = 1.0\), and \(x_3 = 1.5\). Note that the left Riemann sum uses the function values at these points.
3. Evaluate the function \(f(x) = x^2 + 2\) at each left endpoint: calculate \(f(x_0)\), \(f(x_1)\), \(f(x_2)\), and \(f(x_3)\).
4. Multiply each function value by the width \(\Delta x\) to find the area of each rectangle: \(f(x_i) \times \Delta x\) for \(i = 0, 1, 2, 3\).
5. Sum all these rectangle areas to approximate the total area under the curve on [0, 2]: \(\text{Left Riemann Sum} = \sum_{i=0}^{3} f(x_i) \Delta x\). This sum represents the approximate area bounded by the graph and the x-axis.

Key Concepts

Riemann Sums

Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles formed using function values at specific points. The left Riemann sum uses the left endpoint of each subinterval to determine the rectangle height, providing an estimate of the integral.

Partitioning the Interval

Partitioning involves dividing the interval [0, 2] into equal subintervals, here n = 4, to create smaller segments for approximation. Each subinterval has length Δx = (b - a)/n, which is essential for calculating the width of rectangles in the Riemann sum.

Function Evaluation at Endpoints

To compute the left Riemann sum, evaluate the function f(x) = x² + 2 at the left endpoints of each subinterval. These values determine the heights of the rectangles, which when multiplied by the subinterval width and summed, approximate the total area under the curve.