The last few weeks in Calculus have got me so crazy it's unbelievable. I feel like i'm not retaining any information I'm taught. But here we go..
This week we learned about trig inverse integration and volume by disks and washers.
In class I never caught on to trig inverse integration. I know that it's supposedly easy if you recognize the three formulas, but i'm just not getting it. The three formulas are:
1) 1/du arcsin u/a + C
2) 1/dua arctan u/a + C
3) 1/dua arcsec absolute value of u/a + C
Now on to volume by disks.
First off, what is a disk? It's a solid object! I actually understood this in class :)
The formula is pi S[r(x)]^2dx
First you have to sketch the graph. Just plug into your y equals in your calculator and sketch! Depending on what axis it tells you to do the problem on is how you reflect the graph. Once you sketch it, shade the graph and then start working the problem!
Ex: Find the volume of the solid formed by revolving the region bounded by the graph of f(x) = square root of x and on y=0 x=3 about the x-axis.
First you would sketch the graph, which looks like a curved line coming out of (0,0) heading to the right to x=3. You then reflect that about the x axis and shade, which alternatively gives you a shape that looks like half of a football. Now to start the problem.
Pi S(0 - 3) (square root of x)^2 dx
Pi S(0 - 3) x dx
Pi[1/2x^2] 0 to 3
=9pi/2
Volume by washers is the same as volume by disks except it has a hole in it. That means it is bounded by two graphs. The formula for volume by washers is Pi S top^2 - bottom^2 dx.
This also i understood in class.
One thing i still don't understand is MRAM and TRAM, so if anyone can help me with that please do.
The packet we got in class friday i understood and i was actually getting them right, but of course at home i get lost and am not becoming successful with this packet. It's like once i get out of the room, all my knowledge is lost.
Also, we got to try out turning point this week. I got all of the answers right the first day, but it all went downhill from there. Hopefully this will help me to remember everything we've learned before.
Can't wait for another overwhelming week of calculus, sikeeeee.
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MRAM: formula is delta x [ f(mid) + f(mid) + ... ]
ReplyDeleteTrapezoidal formula is
delta x/2 [ f(a) + 2f(a+delta x) + 2f ( a+2 delta x) + ... f(b)]
Its easier to learn by example so:
x^2 - 1 on [0, 4] with 4 subintervals
First step is to find your delta x and you do that by using the formula b-a/n: 4-0/4 = 1
From there, for MRAM you have to find your midpoints. You do that by starting with a and then adding delta x until you get to b
0, 1, 2, 3, 4
then you find the midpoints of those numbers
0+1 / 2 = 1/2
1+2/2 = 3/2
2+ 3 / 2 = 5/2
3+4/ 2 = 7/2
From there all you do is plug into the formula
1 [ f(1/2) + f(3/2) +f(5/2) + f(7/2)]
and solve: 1[ -3/4 + 5/4 + 21/4 + 45/4] which equals 17
Trapezoidal: delta x/2 would equal 1/2
and from there just plug in
1/2 [f(0) + 2f(1) + 2f(2) + 2f(3) + f(4)]
1/2 [-1 + 0 + 6 + 16 + 15] = 1/2 [36] which equals 18
The main part is just remember the formula and which formula goes with each.