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A Riemann sum is a method to approximate the integral of a function over an interval. It's essentially adding up a lot of little rectangles underneath a curve to estimate the total area under the curve. Each rectangle represents a subinterval with a specific width, known as \( \Delta t \). For each subinterval, there is a sample point, \( t_i \), where the function is evaluated. This approach is fundamental in calculus because it gives us a straightforward, yet powerful, way to explore the concept of integrating functions over a continuous range.

• The width of each rectangle is \( \Delta t \), representing tiny pieces of the interval.
• \( f(t_i) \) is the height of the rectangle, given by the function value at the sample point \( t_i \).

By refining the Riemann sum, i.e., making \( n \) larger, the approximation becomes more accurate, approaching the value of the definite integral.

The limit of integration is closely related to the endpoints between which we want to find the total area under a curve by integrating a function. It's important to determine these boundaries correctly to accurately represent the integral as a function of those limits. The integration limit is found from the sample points \( t_i \), calculated based on the given intervals and the number of rectangles (or partitions) in the sum.

• The starting point of integration corresponds to \( t_i \) when the index \( i \) is 0.
• The endpoint correlates with \( t_i \) when \( i \) increases to \( n-1 \), especially as \( n \rightarrow \infty \).

In our example, the integration begins at 1 and ends at 2, as determined by the setup \( t_i = 1 + \frac{i}{n} \). When \( i \) starts at 0, \( t_i = 1 \), and when \( i \) reaches \( n-1 \), \( t_i \) approaches 2. Thus, the limit of integration goes from 1 to 2, producing a clear boundary for the area calculation.

Function evaluation finds the specific value of a function at a given point inside the interval we are focusing on. In a Riemann sum, you use function evaluation to calculate the height of the rectangles mentioned above. When you evaluate the function at each sample point \( t_i \), you determine the contribution of that particular segment to the overall integral.In our case, the function we evaluated is \( f(t_i) = e^{t_i} \). This formula calculates the height of each rectangle at the sample point \( t_i \), which in turn is used to approximate the area under the curve of \( e^t \) over the interval from 1 to 2. Function evaluation helps us capture the change in function values across the interval which is crucial for accurate integration.

Definite integral notation provides a compact and precise way to express the sum of infinite rectangles under a curve as the interval approaches a continuous scale. It's usually written as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration, \( f(x) \) is the function being integrated, and \( dx \) represents an infinitesimal change in the variable \( x \).For our Riemann sum example, the definite integral that represents this sum is \( \int_{1}^{2} e^t \, dt \). By integrating \( e^t \) from 1 to 2, we find the exact total area under the curve from the point where \( t = 1 \) to when \( t = 2 \). This notation makes it clear what function is being integrated and the specific region we are considering, providing a clear picture of the task at hand.