Riemann Sum and Area Approximation

Riemann sums are a method for approximating the area under the curve of a function on a given interval. They work by dividing the area into a series of thin rectangles, calculating the area of each rectangle, and summing those areas. The height of each rectangle is determined by the function's value at a specific point within that rectangle's base (subinterval).

The process involves these steps:

1. **Divide the interval:** Split the interval [a, b] on the x-axis into 'n' smaller subintervals.
2. **Determine the width:** Each subinterval has a width, usually denoted by Δx = (b - a) ÷ n (for equal width subintervals).
3. **Choose a point in each subinterval:** Select a specific point (often called the "sample point" or "evaluation point") within each subinterval.
4. **Determine the height:** The height of the rectangle for that subinterval is the function's value at the chosen point, f(xi*) (where x_i* is the chosen point in the i-th subinterval).
5. **Calculate rectangle areas:** The area of each rectangle is height × width = f(xi*) * Δx.
6. **Sum the areas:** Add up the areas of all 'n' rectangles to get the total approximation:
   Approximate Area ≈ Σi=1n f(xi*) Δx

The types of Riemann sums are determined by how the sample point is chosen within each subinterval:

• **Left Riemann Sum (LRAM):** Uses the left endpoint of each subinterval to determine the rectangle height.
• **Right Riemann Sum (RRAM):** Uses the right endpoint of each subinterval to determine the rectangle height.
• **Midpoint Riemann Sum (MRAM):** Uses the midpoint of each subinterval to determine the rectangle height. This often provides a better approximation.
• **Trapezoidal Rule:** While not strictly a Riemann sum (it uses trapezoids instead of rectangles), it's often taught alongside Riemann sums as another method for approximating area, averaging the left and right endpoint heights for each subinterval.

To get a more accurate approximation of the area under the curve using Riemann sums, you need to **increase the number of subintervals (n)**. As 'n' increases, the width of each rectangle (Δx) decreases, and the tops of the rectangles fit the curve more closely, reducing the error in the approximation.

In fact, the definite integral, which gives the *exact* area under the curve, is defined as the limit of a Riemann sum as the number of subintervals approaches infinity:

∫_a^b f(x) dx = lim_n→∞ Σ_i=1ⁿ f(x_i*) Δx

No. Riemann sums can approximate the "net area" under a curve, even if the function takes on negative values. When the function is negative in a subinterval, f(xi*) will be negative, resulting in a negative area for that rectangle. The sum of these signed areas gives the net area (area above the x-axis minus area below the x-axis).