Riemann Sums What is a Riemann Sum Takes
- Slides: 16
Riemann Sums
What is a Riemann Sum? Takes approximating the area under a curve with rectangles to the next level LOOKS LIKE: Adds the rectangles, where n is the number of partitions (rectangles) Width of partition: Height of rectangle for each of the x -values in the interval
What Do We Do With This? If the number of partitions is allowed to approach infinity…what happens? That’s right! The rectangular approximation approaches the EXACT area under the curve! How do we do it?
That Limit Equals Something Special!
What in the World? READ LIKE THIS: The integral, from a to b, of f(x) with respect to x
What’s With This Thing? Upper (right) limit of integration Integrand (between the integral sign and the dx) Integral sign (originated from the summation sign, Sigma) Lower (left) limit of integration Tells you the variable of integration (who is the variable)
Examples of How to Read The integral, from 1 to 4, of x 2 with respect to x is 21
Writing Riemann Sums as Integrals Where x is between 0 and 2.
Writing Riemann Sums as Integrals Where x is between 3 and 7.
Writing Riemann Sums as Integrals Where x is between -2 and 2.
Definite Integrals Why is it called a definite integral? Because the integral sign includes limits of integration!
Properties of Definite Integrals
Properties of Definite Integrals
Properties of Definite Integrals Why? • Because the integral is asking for the area under the curve between 2 and 2…there is no area covered!
Properties of Definite Integrals Why is this okay? • Because an integral is just a limit of a Riemann Sum…so all the properties of limits hold for integrals!
Properties of Definite Integrals