so this week we learned all kinds of new stuff, and we practiced using the little clicker things to take quizzes... so, here are some things we learned:
washers are like disks, but they have holes in them. washers has two equations, volume of washers is S top^2-bottom^2 dx, and area of washers is S top-bottom. to do washers, all you have to do is first draw the graphs of the two equations, then you use whichever formula applies, either for volume or area, then you set them equal and solve for x to find the bounds. then you plug in the bounds and then integrate.
a disk is a solid object, and to find the volume of it, you just solve for x or y of the given equation. you can tell by the problem which axis its about. so you draw the equation, then you reflect it. the formula is S r^2 dx, and the radius is the equation that is given. and the formula for the area of a disk is the same thing but the radius is not squared.
another thing we learned was trig inverse integration, and the three formulas for it are:
S du/(sqrta^2-u^2)=arcsin u/a +c
S du/(a^2+u^2)= du(1/a) arctan u/a+c
s du/(usqrtu^2-a^2)= 1/a arcsec lul^e/a+c
things I don't understand: to be honest I hardly understand any of the stuff I just mentioned, I have almost no idea how to do anything we learned in like the past three weeks... so everything from like basic integration until now I do not get. And I don't know what to do about it..
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I completely feel your pain!!! I had NO IDEA how to do any of that stuff until Malerie stayed up with me until 4 in the morning trying to teach me Calculus
ReplyDeleteI will admit, I went rabid a few times and almost ate (literally) the packet we had to do...but anyway
I can help with integration on here...I can't really explain anything else over the computer
Step 1. add one to the exponents
Step 2. divide the coeff by the new exponent
That's how you get the equation. If it's indefinite, add C. If it's definite, plug in the bottom number minus the top number =]=]
Disks and washers are extremely easy once it clicks.
ReplyDeleteremember that a disk is a solid object, it will only have one equation.
formula is pi S[r(x)]^2dx .. r is your radius/your gived equation.
all you do is plug the equation in, integrate it and then plug in the two points that are givin, don't for get you distribute the pi into your final answer
for ln integration most of the time... the top of the fraction will be the derivative of the bottom. For instance..lets say you have 4/4x+2..this is ln integration. your u would be 4x+2 and your du would be 4. So your final answer would be ln|4x+2|+C..
ReplyDeletedisks and washers are easier than they look. disks are solid objects, washers are objects with holes in them (they will have two equations). you mostly just need to remember the equations for these two, which are not hard at all! volume is squared, area is not! for both disks and washers, same equation, just not squared when it asks for area. if it asks for volume, always put a pi in front of the integral. if it asks for area, you don't put pi.
ReplyDeleteformula for volume of a disk
pi S r(x)^2dx
formula for area of a disk
S r(x)dx
r=radius. radius is your given equation. it's that easy. so pretty much plug in the equation into your formula, integrate it, and solve.
formula for volume of a washer
pi S top^2 - bottom^2 dx
formula for area of a washer
S top-bottom dx
you are given two equations.. plug both into your calculator and see which equation is on top and which is on bottom on the graph that is given.
hope i helped :)
volume by disks:
ReplyDeletethe formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
volume by washers:
the formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.