Finding exact area with the Definite Integral

Okay i was busy trying to figure these things out and then i got stuck.

Now using the Riemann notation to calculate the area under the graph we get the following:

(I calculated )

They use rectangles under the graph to approximate the area. (6 rectangles and the x-axis value ranges from 0 to 3.)

Then the book goes further and says:

For any number of rectangles under the graph(Still ranging from 0 to 3 on the x-axis):

At the end we arrive at the following:

Up to there i understand perfectly.

Then from that we get

Now i have no idea why Shouldn't it just equal ?

And now we arrive at integration(Still for the graph ):

And the second question: Do we always have to use a Riemann sum to arrive at this part: ? What, no shortcut? :D

And could a mod please delete this(Posted by accident): http://www.mathhelpforum.com/math-he...html#post85119