Definite integral as the limit of a Riemann sum | AP Calculus AB | Khan Academy

TL;DR

Riemann sums are used to approximate the area under a curve by dividing it into rectangles, and the Riemann integral gives a rigorous definition of the area under a curve.

Transcript

We've done several videos already where we're approximating the area under a curve by breaking up that area into rectangles and then finding the sum of the areas of those rectangles as an approximation. And this was actually the first example that we looked at where each of the rectangles had an equal width. So we equally partitioned the interval b... Read More

Key Insights

• 🍹 Riemann sums approximate the area under a curve by dividing it into rectangles.
• ↔️ The height of the rectangles can be determined in different ways, such as using left or right endpoints, midpoints, or trapezoids.
• 🥡 The Riemann integral provides a rigorous definition of the area under a curve by taking the limit of Riemann sums as the number of rectangles increases.
• 💦 Bernhard Riemann is known for his contributions to mathematics, particularly for his work on Riemann sums and the Riemann integral.
• 🍹 The Riemann integral represents the exact area under a curve, while Riemann sums provide approximations.
• 🥺 Increasing the number of rectangles in a Riemann sum leads to a more accurate approximation of the area under the curve.
• 💱 Delta x represents the width of each rectangle in a Riemann sum, and dx represents an infinitely small change in x.

Questions & Answers

Q: What are Riemann sums and how are they used to approximate the area under a curve?

Riemann sums are approximations of the area under a curve by dividing it into rectangles and summing their areas. By increasing the number of rectangles, we can get a better approximation of the actual area.

Q: What are the different ways to determine the height of the rectangles in Riemann sums?

The height of the rectangles can be determined by using the function evaluated at the left endpoint, right endpoint, midpoint, or by using trapezoids. Each method provides a slightly different approximation of the area under the curve.

Q: Who is Bernhard Riemann and what is he known for?

Bernhard Riemann was a mathematician who made significant contributions to mathematics. He is most known for his work on Riemann sums, which are used to define the Riemann integral, the most mainstream and rigorous definition of an integral.

Q: How is the Riemann integral calculated?

The Riemann integral is calculated by taking the limit of Riemann sums as the number of rectangles approaches infinity. The rectangles become better and better approximations of the actual area under the curve, and the integral represents the exact area.

Summary & Key Takeaways

• Riemann sums approximate the area under a curve by dividing it into rectangles and summing their areas.

• There are different ways to determine the height of the rectangles, such as using the left, right, or midpoint endpoints, or even using trapezoids.

• The Riemann integral provides a rigorous definition of the area under a curve by taking the limit of Riemann sums as the number of rectangles approaches infinity.