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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

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

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The integral, from a to b, of f(x) with respect to x

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)

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

Where x is between 0 and 2.

Where x is between 3 and 7.

Where x is between -2 and 2.

Definite Integrals

Why is it called a definite integral?

Because the integral sign includes limits of integration!

Why?

- Because the integral is asking for the area under the curve between 2 and 2…there is no area covered!

Why is this okay?

- Because an integral is just a limit of a Riemann Sum…so all the properties of limits hold for integrals!