Defining integrals with Riemann sums

Here are the key points to learn when studying "Definite Integral as the Limit of a Riemann Sum":

1. Definition of Riemann Sum: A method to approximate the area under a curve by dividing the interval into subintervals, calculating the area of rectangles formed, and summing these areas.

2. Subdivision of Interval: The interval $[a, b]$ is divided into $n$ subintervals of equal width $\Delta x$, where $\Delta x = \frac{b - a}{n}$.

3. Sample Points: Choose sample points $x_i^*$ within each subinterval to evaluate the height of the rectangles. Common choices include left endpoints, right endpoints, or midpoints.

4. Riemann Sum Formula: The Riemann sum is expressed as:

   $$S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x$$

5. Limit as $n$ Approaches Infinity: The definite integral is defined as the limit of the Riemann sum as the number of subintervals $n$ approaches infinity:

   $$\int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} S_n$$

6. Fundamental Theorem of Calculus: Establishes the connection between differentiation and integration, stating that if $F$ is an antiderivative of $f$, then:

   $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

7. Continuity and Riemann Integrability: A function that is continuous on $[a, b]$ is Riemann integrable, and the limit of the Riemann sums converges to the definite integral.

8. Practical Application: Use Riemann sums to estimate definite integrals, leading to more insight into the behavior of functions over intervals.

Understanding these points lays the groundwork for a deeper comprehension of integral calculus and its applications.

When studying "Worked example: Rewriting definite integral as limit of Riemann sum," focus on the following key points:

1. Understanding Definite Integrals: Recognize that a definite integral represents the area under a curve over a specified interval.

2. Riemann Sums: Learn how Riemann sums approximate the area under a curve by dividing the interval into smaller subintervals and summing the areas of rectangles formed.

3. Limit Process: Understand that as the number of subintervals increases (and their width decreases), the Riemann sum converges to the exact value of the definite integral.

4. Formulas and Notation: Familiarize yourself with the notation for Riemann sums (using delta $x$, partition points) and how it relates to the definite integral.

5. Application: Practice rewriting a definite integral as a limit of a Riemann sum by determining the appropriate function, interval, and partition.

6. The Fundamental Theorem of Calculus: Recognize the connection between the process of integration and differentiation, and how this theorem underpins the relationship between Riemann sums and definite integrals.

By mastering these points, you'll gain a solid understanding of how to relate definite integrals to Riemann sums.

When studying "Worked example: Rewriting limit of Riemann sum as definite integral," focus on the following key points:

1. Definition of Riemann Sum: Understand what a Riemann sum is, typically represented as a sum of function values times the width of subintervals.

2. Limit Process: Recognize that a limit of a Riemann sum as the number of subintervals approaches infinity leads to the definite integral.

3. Formulating the Integral: Learn how to express the Riemann sum in terms of a definite integral, identifying the interval of integration and the function being integrated.

4. The Fundamental Theorem of Calculus: Understand how the concept of limits ties into the evaluation of definite integrals using antiderivatives.

5. Applying the Concept: Practice rewriting various Riemann sums as definite integrals, ensuring proficiency in transitioning between the two concepts.

By focusing on these aspects, you can gain a deeper understanding of the relationships between Riemann sums and definite integrals in calculus.