okay so this week was my first week back in two weeks, ehhh i was lost
thankfully, i got extremely excited friday bc i finally had an idea of what i was doing with those packets! and guess what... i acutally finished a packet all by myself!
This week we learned about inverse integration and volume by disks and washers.
i think that the volume by disks and washers are the probably the two concepts i have caught on to the quickest so far this year.
THE DIFFERENCE BETWEEN A DISK AND A WASHER: a disk is a solid object, circle..
a washer has a hole/holes in it, kind of like the gameee!
FORMULA FOR DISKS: pi S[r(x)]^2dx, this formula is almost exactly the same as when you are asked to find the area of a disk, however, while finding the area you do not square the radius (the given equation).
FORMULA FOR WASHERS: pi S[(top)^2-(bottom)^2]dx
EXAMPLE: find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 4. y=sq root of -x^2+6x+27
4
pi S (sq root of -x^2+6x+27)^2dx
0
pi[-1/3X^3+3X^2+27X]= pi [0-(404/3)]= 404/3 pi
it is not necessary for you to graph the equation when you are doing things involved with disks, but when doing washers you MUST graph both of the equations to see which one's the top equation and which one is the bottom equation.
Some things i still don't understand are the trig invers stuff and i know that when i see a fraction and it asks me to integrate it i DON'T USE QUOTIENT RULE, that's stuck in my headdddd but i dont know what to do if i dont do that, i forgot. Help would really be appreciated, thanks guys. ohh and the formula for the trapezoidal rule, i lostt it.
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when you are given a fraction you can either raise the bottom to -1 and multiply it by the top, or you can use ln integration.
ReplyDeletenatural log integration actually isn't as hard as it seems. msot of the time, the top of the fraction is the derivative of the bottom of your fraction, just like trina said, ex: 4/4x.. but sometimes you have to substitue in something.. like you were given 1/4x, you would need to substitute in a 4 because the derivative of 4x is 4, not 1. so your u=4x du=4dx, and you would simply but a 1/4 in front of the integration symbol. this is ln integration.
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ReplyDeleteln integration is very very very simple. you just need to recognize or identify it. after that, it becomes all easy.
ReplyDeleteFor instance, when you have 1 over x. you bring the x to the top, it's not x to the negative one. -1 + 1 = 0 so we can't do this. This is when you know you have a ln integration.
The integral of 1/x is ln x.
So the integral of 5/x is 5lnx
Integral of 6/(x+1) is 6ln|x+1| and so on...