The following functions are positive and negative on the given interval.
ƒ(𝓍) = xe⁻ˣ on [-1,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Verified step by step guidance

1. Step 1: Divide the interval [-1, 1] into n = 4 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (1 - (-1)) / 4 = 2 / 4 = 0.5.
2. Step 2: Identify the endpoints of the subintervals. The subintervals are [-1, -0.5], [-0.5, 0], [0, 0.5], and [0.5, 1].
3. Step 3: For the left Riemann sum, use the left endpoint of each subinterval to evaluate the function ƒ(𝓍). The sum is given by: Σ (Δ𝓍 * ƒ(left endpoint)).
4. Step 4: For the right Riemann sum, use the right endpoint of each subinterval to evaluate the function ƒ(𝓍). The sum is given by: Σ (Δ𝓍 * ƒ(right endpoint)).
5. Step 5: For the midpoint Riemann sum, use the midpoint of each subinterval to evaluate the function ƒ(𝓍). The sum is given by: Σ (Δ𝓍 * ƒ(midpoint)).

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's value at that point is multiplied by the width of the subinterval. The sum of these products gives an approximation of the total area. There are different types of Riemann sums, including left, right, and midpoint sums, which differ based on the choice of sample points.

Left, Right, and Midpoint Riemann Sums

In a left Riemann sum, the function's value at the left endpoint of each subinterval is used to calculate the area. Conversely, a right Riemann sum uses the right endpoint. A midpoint Riemann sum takes the function's value at the midpoint of each subinterval. Each method provides a different approximation of the area, and the choice of method can affect the accuracy of the approximation.

Net Area

The net area refers to the total area between the graph of a function and the x-axis over a specified interval, accounting for areas above the x-axis as positive and areas below as negative. In the context of Riemann sums, calculating the net area involves summing the contributions from both positive and negative sections of the function within the interval. This concept is crucial for understanding how the function behaves over the interval and how it affects the overall area calculation.