Help! Area in Integration (I ask this before but no one bothers)?

Question

Area in integration?

Pls show me the correct answer, and how to get the measurement of length this is the image, thanks very much..

the image is just draw in ms paint pls dont blame me

http://img223.imageshack.us/my.php?image=integrationet0.jpg

Pls show me the precise way, because I want to learn something new!

Answer 1

I think the area under the curve is just int (function) dx

dx is simple, if your figure is correct it simply goes from -2 to 2

The area above the curve is the area of the rectangle - the area under.

I think that you do not need any complicated length.

Maybe I missed the point. :)

Answer 2

(Read this with the page you linked in another window so you can reference it).

That little white rectangle in the upper left of the figure is the clutch dude.  It has a width of dx, and a height of

4 - (4x - x^2)  

{That is what you were calling "length".  That is the top curve minus the bottom curve.  Note that it still works when the bottom curve goes negative, as people have an issue seeing that at first for some reason}

So what we want to do to find the area of the entire purple guy is sum up a bunch of such little white rectangles.  How we do that is we start at the left (where x = -2) and chop up the entire guy into rectangles of width dx until we get to the right (where x = 2).  That gives us our upper and lower limits of integration.  Now all we do is sum up all the little rectangles (that is do the integration)

Integrate[4 - (4x - x^2) dx ,{ from x = -2 to x = 2}]

I'll let you finish the rest, it is fairly straightforward.