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Riemann sums are used to approximate the definite integral of a function, which represents the area under the curve. They provide a foundational understanding of integration before moving on to more sophisticated techniques.

Types of Riemann Sums

There are three main types of Riemann sums, each differing in how the rectangle heights are determined:

1. Left Riemann Sum: The height of each rectangle is determined by the function’s value at the left endpoint of the subinterval.

2. Right Riemann Sum: The height of each rectangle is determined by the function’s value at the right endpoint of the subinterval.

3. Midpoint Riemann Sum: The height of each rectangle is determined by the function’s value at the midpoint of the subinterval.

Calculating Riemann Sums

Let’s consider a function $ f(x) $ on the interval $ [a, b]] $ . We divide this interval into $ n $ subintervals of equal width, $ \Delta x = \frac{b-a}{n} $ . The $ i $ -th subinterval is $ [x_{i-1}, x_i]] $ , where $ x_i = a + i\Delta x $ .

The general formula for a Riemann sum is:

$$ \sum_{i=1}^{n} f(x_i^) \Delta x $$ where $ x_i^ $ is a point in the $ i $ -th subinterval. The choice of $ x_i^* $ determines the type of Riemann sum:

• Left Riemann Sum:

$$ x_i^* = x_{i-1} = a + (i-1)\Delta x $$ * Right Riemann Sum:

$$ x_i^* = x_i = a + i\Delta x $$ * Midpoint Riemann Sum:

$$ x_i^* = \frac{x_{i-1} + x_i}{2} = a + (i - \frac{1}{2})\Delta x $$ ### Example: Approximate the area under the curve $ f(x) = x^2 $ from $ x=0 $ to $ x=2 $ using a right Riemann sum with $ n=4 $ subintervals.

$ \Delta x = \frac{2-0}{4} = 0.5 $

The subintervals are $ [0, 0.5 $ , $ [0.5, 1 $ , $ 1, 1.5 $ , $ 1.5, 2 $ .

The right Riemann sum is: $ R_4 = f(0.5)(0.5) + f(1)(0.5) + f(1.5)(0.5) + f(2)(0.5) = 0.5(0.5^2 + 1^2 + 1.5^2 + 2^2) = 0.5(0.25 + 1 + 2.25 + 4) = 3.75 $

y = x^2
y = 0
x = 0
x = 2

Error Analysis

The error in a Riemann sum approximation is the difference between the actual value of the definite integral and the approximation. The error decreases as the number of subintervals ( $ n $ ) increases. Generally, the error is proportional to $ \frac{1}{n} $ . More sophisticated error bounds exist, but are beyond the scope of AB calculus.

Relationship to Definite integrals

As the number of subintervals ( $ n $ ) approaches infinity, the Riemann sum converges to the definite integral:

$ \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x = \int_a^b f(x) dx $

This is the fundamental theorem of calculus, connecting the concept of Riemann sums to the process of integration.

Applications of Riemann Sums

Riemann sums are not just theoretical tools; they have practical applications in various fields, including:

• Approximating areas of irregularly shaped regions: When dealing with complex shapes where standard geometric formulas are not applicable.
• Calculating work done by a variable force: The force may change over the distance it acts upon.
• Estimating the total value of a continuously changing quantity: Such as the accumulation of rainfall over time or the growth of a population.

This rundown provides a comprehensive overview of Riemann sums for Calculus AB. Remember to practice various examples to solidify your understanding.