Ugh, I forgot to post my blog this weekend -.-
Mom's 40th and major planning.
Anyway, Integration, YAY!
I get the simple integration..Just work backwards! =]
For example, say you have (okay, I dunno how to make the cool symbol, so I'm using S)
S(-4x +2)dx and you want to integrate it using indefinite integration.
1. I usually leave a space and tackle my x's first
Since the derivative of __x^2 is 2(__)x, let's write it like this!
___x^2 +___x + C (DON'T FORGET PLUS C!)
2. Take your coefficient and divide by your exponent to get your original coefficient!
(-4/2)x^2+(2/1)x + C
3. Simplify
-2x^2+2x+C
TA DA!
Okay, next: ln Integration *echoes*
Whenever you have a fraction where the top is the derivative of the bottom, you have to deal with ln Integration.
Just take the bottom, put it inside ln| |+C and see what needs to be added.
Example.
S(2/x)dx
1. You can make 2 the derivative of x by multiplying x by 2. (2x)
2S(1/x)dx
2. Now, put the bottom inside ln| |+C and remove the S
2ln|x|+C
THINGS I'M HAVING TROUBLE WITH!!!
1. Can you ALWAYS manipulate a problem to make it solvable by ln Integrations?
Ex: (x^2+x+1)/(x^2+1)
2. How exactly do you do synthetic for ln Integration? I'm a little confused.
On this beast packet:
1. How do you integrate something like these?
a) S(-5/x)dx
b) S -x^-1
c) S x^-1
2. For numbers 16-25, what does "subject to the initial condition...." mean???
3. Can someone PLEASE explain how to do RRAM, LRAM, MRAM, and TRAM???????
4. How do you go about solving number 46? I have no idea where to even begin.
That is all for the moment, if people comment this, I might actually get somewhere ^^
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when you go to do synthetic division, your u is what u are dividing. so if your u is x-2 and you have other numbers to divide, you will use those numbers and plug in for x.
ReplyDeletefor sovling 5/x. i would leave the 5 and make x^-1 and solve from there. Hope this helps you!
LRAM, RRAM, MRAM, and Trapezoidal is basically plugging into the formula for each. I find these easier to understand by using an example instead of just saying what the formula is so..
ReplyDeletef(x) = x^2 - 3 on [1,4] n=3
So the first step in using all four is to find your delta x which is found by using the formula: b-a/n
4-1/3 = 3/3 = 1
LRAM: delta x [ f(first number in interval) + f( 1st # + delta x) + f( that # + delta x)]
1[ f(1) + f(2) + f(3)]
you only go up to three because your n=3
from there you plug into your equation:
1[ (1)^2 - 3 + (2)^2 - 3 + (3)^2 - 3 ] = 1[5] = 5
RRAM: same thing except start with f(# + delta x) and you should end with f(b)
1[ f(2) + f(3) +f(4)]
Then plug in: 1[ 1+ 6 + 13] = 20
MRAM is slightly different because you have to find the slope between each number. The numbers on your interval are: 1, 2, 3, and 4.
1+2 / 2 = 3/2
2+3/2 = 5/2
3+4/ 2 = 7/2
now plug in those numbers: [1 (f(3/2) + f(5/2) + f(7/2)] = 47/4
Trapezoidal: instead of multiplying by delta x, you will multiply by delta x/ 2 and you will also have one extra term then your n so four in this case. You use the numbers given in your interval and multiply the middle numbers by 2.
1/2[ f(1) + 2f(2) + 2f(3) + f(4)]
1/2[ -2 + 2 + 12 + 13] = 25/2
for synthetic division, your u is what you are dividing. so if your u is x-2 and you have other numbers to divide, you will use those numbers and plug in for x.
ReplyDeletefor 5/x. i would leave the 5 and make x^-1 and solve from there.
hope this is right!
Great explanation of Riemann sums Trina.
ReplyDeleteLRAM is left hand approximation and the formula is:
ReplyDeletedelta x [f(a) + f( delta x +a) .... + f( delta x - b)]
Say you are asked to calculate the left Riemann Sum for -4x -5 on the interval [-3, -1] divided into 2 subintervals.
delta x would equal: -1+3 /2 = 2/2 = 1
1[ f(-3) + f(-3 +1)]
1[ f( -3) + f(-2)]
then plug into your equation
RRAM is right hand approximation and the formula is:
delta x [ f(a + delta x) + .... + f(b)]
so using the same example:
1[ f( -2) + f(-1)] and then plug into your equation
MRAM is to calculate the middle and the formula is:
delta x [ f(mid) + f(mid) + .... ]
To find midpoints, you would add the two numbers together then divide by two
In this problem the numbers would be: -3 , -2, -1
-3 + -2/ 2 = -5/2 and -2 + -1 / 2 = -3/2
so 1[f(-5/2) + f(-3/2)] and the plug in
Trapezoidal is different because instead of multiplying by delta x, you multiply by delta x/2 and you also have on more term then your number of subintervals.
The formula is : delta x/2 [f(a) + 2f(a + delta x) + 2f(a+ 2 delta x) + ....f(b)]
For this problem: 1/2 [ f(-3) + 2 f(-2) + f( -1)] and then plug in.