6.1 Areas between curves

OK WELL LET ME START OFF BY SAYING..... BEWARE.... I, LAUREN, AM ON A PC. ENOUGH SAID.

6.1
In this section we will be finding the area between curves that are not necessarily bound by just the x and y axis


Example 1:









Find the area of the bound by f (x) and g (x) on the interval [a,b]







To find the area, all you would need to do is subtract the integral of f (x) on the interval [a,b] from the integral of g (x) on the same interval.



Example #2

Now, find the area bound by the two graphs given below.











Graph it on the calculator to see it





Luckily, we already know that one of the intersection points comes at (0,0). We can use our graphing calculators to find the x-value for the other point. You can go to CALC then to INTERSECT, then magically we find out that the intersection is at the point 1.1807757. Immediately after you find this information out, store the value as the "a" value. Then, with more help form the calculator, you can calculate the area by calculating the integral.

It should look like this on your calculator:

fnInt (Y1 - Y2 , x , 0 , A) = .7853885505


This is because you are caluculating the integral of the difference between the two functions Y1 and Y2, in terms of x, on the interval 0 to A. (You already calculated and stored the value.)



When you have a problem that looks similar to a sinusoid, you have two options:
1. you can add the integrals comprised of one function subtracted from the other on the intervals of the intersection points. So, adding up the segments.


2. you can find the absolute value of the integral of the difference of two funtions, completely disregarding the segments. (Shown below)



EXAMPLE 3



Sometimes, merely subtracting one integral from the other is not the easiest way of doing things. When the rectangles that make the area and integral are horizontal and not vertical, it is easier to put things in terms of y instead of x.



For example, the equations y = x-1 and y^2 = 2x + 6











So, to work horizontally, rework the y equations to x.





x = y + 1


x = .5 y^2 - 3




Now we are left with the integral
























Now you basically know how to find an area between curves. I bet you feel a lot smarter now that you know this. I know I do.



Quick link to check out: http://archives.math.utk.edu/visual.calculus/5/area2curves.3/







Mr. French, here's one for you: http://www.teacherschoice.com.au/Maths_Library/Calculus/area_between_two_curves.htm









A friendly reminder that Grey's Anatomy is back on tomorrow, so you better all watch it!







JAMESYWAMSEY YOU'RE UP NEXT......












man i'm excited already!!!!!!

-Lauren