Riemann sums
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Riemann Sums. Width of partition:. Adds the rectangles, where n is the number of partitions (rectangles). Height of rectangle for each of the x-values in the interval. What is a Riemann Sum?. Takes approximating the area under a curve with rectangles to the next level LOOKS LIKE:.

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Riemann Sums

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Riemann sums

Riemann Sums


What is a riemann sum

Width of partition:

Adds the rectangles, where n is the number of partitions (rectangles)

Height of rectangle for each of the x-values in the interval

What is a Riemann Sum?

  • Takes approximating the area under a curve with rectangles to the next level

    LOOKS LIKE:


What do we do with this

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

That Limit Equals Something Special!


What in the world

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

What’s With This Thing?

Integrand (between the integral sign and the dx)

Upper (right) limit of integration

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

Examples of How to Read

  • The integral, from 1 to 4, of x2 with respect to x is 21


Writing riemann sums as integrals

Writing Riemann Sums as Integrals

Where x is between 0 and 2.


Writing riemann sums as integrals1

Writing Riemann Sums as Integrals

Where x is between 3 and 7.


Writing riemann sums as integrals2

Writing Riemann Sums as Integrals

Where x is between -2 and 2.


Definite integrals

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 integrals1

Properties of Definite Integrals


Properties of definite integrals2

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 integrals3

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 integrals4

Properties of Definite Integrals


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