Post 2: Riemann Sums and Approximations

Riemann sums are a method used in calculus to approximate the area under a curve. This method involves dividing the region under the curve into smaller rectangles and summing their individual areas to get an approximation of the total area.

There are several types of Riemann sums, including left, right, and midpoint sums. The choice of which type to use depends on how the rectangles are positioned relative to the curve.

1. Left Riemann Sum: In a left Riemann sum, the left endpoints of the rectangles touch the curve. The width of each rectangle is determined by the division of the total interval into equal subdivisions, called partitions. The formula for the left Riemann sum is:

Where:

• $L_n$ represents the left Riemann sum.
• $f(x_i)$ represents the value of the function at the left endpoint of each subdivision.
• $\Delta x$ represents the width of each rectangle, which is equal to $\frac{b-a}{n}$, where $a$ and $b$ are the lower and upper limits of the interval, respectively, and $n$ is the number of subdivisions.

2. Right Riemann Sum: In a right Riemann sum, the right endpoints of the rectangles touch the curve. The formula for the right Riemann sum is similar to that of the left Riemann sum, but the function values are evaluated at the right endpoints instead.

Where:

• $R_n$ represents the right Riemann sum.
• $f(x_i)$ represents the value of the function at the right endpoint of each subdivision.

3. Midpoint Riemann Sum: In a midpoint Riemann sum, the midpoints of the rectangles touch the curve. The formula for the midpoint Riemann sum is:

Where:

• $M_n$ represents the midpoint Riemann sum.
• $f(\bar{x}_i)$ represents the value of the function at the midpoint of each subdivision.

Now, let's look at an example using the left Riemann sum. Consider the function $f(x) = x^2$ on the interval $[0, 4]$ and we want to approximate the area under the curve using 4 subdivisions.

1. Determine the width of each rectangle: $\Delta x = \frac{4 - 0}{4} = 1$

2. Evaluate the left endpoint function values: $f(x_0) = f(0) = 0^2 = 0$ $f(x_1) = f(1) = 1^2 = 1$ $f(x_2) = f(2) = 2^2 = 4$ $f(x_3) = f(3) = 3^2 = 9$ $f(x_4) = f(4) = 4^2 = 16$

3. Calculate the left Riemann sum: $L_4 = 0 \cdot 1 + 1\cdot 1 + 4\cdot 1 + 9\cdot 1 + 16\cdot 1 = 30$

Thus, the left Riemann sum of $f(x) = x^2$ on the interval $[0, 4]$ with 4 subdivisions is equal to 30, approximating the area under the curve. Similarly, the right and midpoint Riemann sums can be calculated using the respective formulas.