Wednesday, January 10, 2007

6.1 Areas between curves


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)


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:

Mr. French, here's one for you:

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


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



At 11:50 PM, Blogger jamesy wamesy said...

Hi, Lauren. Good blog. I especially like the part where you spell function "funtion," or when you say "after you find this information out," leaving a dangling preposition.

At 6:54 AM, Blogger Math Maverick said...

In light of James' comment:

Dear Dr. Science,

Is there a reason for not ending a sentence with a
preposition that you can think of?

from John Mostrom of Seattle, WA

I must admit I don't know where you're coming from.
Correct usage in English and Science is something
I've devoted my whole life to. Of course, if I say
anything you can't understand, it will just become
a new hammer you can try to hit me or another expert
over the head with. There are plenty of people like
you I can't hope to change the mind of. But then,
I've dealt with people like you before. People who
don't really want to learn, but just hope to find
someone they can publicly disagree with. There's
little I can say that your type won't find something
to object to. But getting back to your question,
no there's really no reason for not ending a sentence
with a preposition, at least none I can think of.

-Ask Dr. Science
22 Nov 2006

At 9:39 PM, Blogger Isabella said...

Wow. I was just tremendously amused by Dr. Science. Something may or may not need to be done about my grammar fetish. ~8)

At 12:52 AM, Blogger Taylor said...

Oh, Dr. Science, you've done it again, you rascal.


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