Finding The Area Using The Limit Definition & Sigma Notation

TL;DR

This video explains how to find the area under a curve using the limit definition, specifically focusing on the right endpoints method.

Transcript

in this video we're going to focus on finding the area using the limit definition so let's say if we have the function x cube and we want to find the area of the shaded region between the x-axis and the curve over the closed interval from zero to eight so let's draw a picture so x cube on the right side looks like this and so from 0 to 8 we want to... Read More

Key Insights

• 🛩️ The area under a curve can be approximated by dividing the interval into smaller rectangles.
• 👻 The limit definition allows us to find the exact area by increasing the number of rectangles infinitely.
• 👈 The right endpoints method is simpler than the left endpoints method for finding the value of x sub i.

Questions & Answers

Q: What is the formula for finding the area under a curve using the limit definition?

The formula is the limit as n approaches infinity of the sum from 1 to n of f(x sub i) times delta x. It involves dividing the interval into smaller rectangles and adding up their areas.

Q: How do you determine the value of x sub i using the right endpoints method?

Using the right endpoints method, x sub i is equal to a plus delta x times i, where a is the beginning of the interval, delta x is the width of each rectangle, and i is the subscript of the rectangle.

Q: Can the left endpoints method be used instead of the right endpoints method?

Yes, both methods can be used to find the area under a curve, but the right endpoints method is often simpler. The left endpoints method uses x sub i = a + delta x times (i - 1).

Q: How is delta x calculated?

Delta x is calculated by dividing the length of the interval (b - a) by the number of rectangles (n). For example, if the interval is from 0 to 8 and n is 4, delta x is 8/4 = 2.

Summary & Key Takeaways

• The area under a curve can be found using the limit definition and the formula: the limit as n approaches infinity of the sum of rectangles from 1 to n of f(x sub i) times delta x.

• The value of x sub i depends on whether the right or left endpoints method is used, but for this example, the right endpoints method is demonstrated.

• Delta x represents the width of each rectangle and is calculated by dividing the interval length by the number of rectangles.