Riemann Sums, Formula, Using Calculator, Examples, Practice Problems - Calculus

A Riemann sum approximates the area under a curve by dividing the region into a series of rectangles and summing their areas. The width of each rectangle is a subinterval, and its height is determined by evaluating the function at a specific point within that subinterval, such as the left endpoint, right endpoint, or midpoint. By increasing the number of rectangles, the Riemann sum becomes a more accurate approximation of the exact area under the curve, which is found by taking the limit as the number of rectangles approaches infinity.

💡How Riemann Sums Work
• Divide the Interval: The interval over which you want to find the area is divided into smaller, equally-sized subintervals.
• Construct Rectangles: For each subinterval, a rectangle is formed with the x-axis as its base.
• Determine Rectangle Height: The height of each rectangle is found by evaluating the function at a chosen point within its subinterval. Common choices include:
◦ Left Endpoint Rule: The height is determined by the function's value at the left-most point of the subinterval.
◦ Right Endpoint Rule: The height is determined by the function's value at the right-most point of the subinterval.
◦ Midpoint Rule: The height is determined by the function's value at the midpoint of the subinterval.
• Calculate Rectangle Area: The area of each individual rectangle is calculated by multiplying its width (the subinterval's width) by its height.
• Sum the Areas: The Riemann sum is the total of the areas of all these rectangles.

💡The Relationship to Integration
• A Riemann sum is an approximation of the definite integral. As the number of rectangles increases, the approximation becomes more precise. When the number of rectangles approaches infinity, the Riemann sum converges to the exact area under the curve, which is the value of the definite integral.

💡Purpose of Riemann Sums
• Approximation: Riemann sums provide a way to estimate the area under a curve when an exact antiderivative might be difficult or impossible to find.
• Foundation of Integration: They serve as a fundamental concept in calculus, illustrating how a continuous function's area can be broken down and added up.
• Building Intuition: By visualizing the process of subdivision, approximation, and summation, Riemann sums help in understanding the underlying principles of integral calculus.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Intro to Riemann sums
01:26 Left Riemann sum
02:42 Right Riemann sum
03:33 Midpoint Riemann sum
04:47 Trapezoidal sum
06:20 Worked examples

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