Okay so this week .FOR ME. was tough!
First we learned volume by disks
so you know that the formula is pi times the integral of the [function given] squared times dx. well then you gotta know what to do right 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...oh, REMEMBER TO GRAPH..
just look at what they give you...they're should be numbers that they want them inbetween
IF NOT..WHAT DO YOU DO???
Next we learned volume by washers
so you know that 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 inbetween 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. REMEMBER TO GRAPH!
just a reminder if it's for the x-axis then you solve for y and if it's on the y-axis then you solve for x..and if you plug the y-axis one's in your calculator to graph then you'll have to turn your calculator sideways with the screen on the right to see how it would really look.
we also learned about area...it's worked the same just they'll give you two equations and if you don't have numbers for the inbetween then you'll set them equal to get them, but if you do then don't worry and keep going. for area there is no pi and no squaring. so you put the integral of the first equation minus the second equation and simplify and solve. then take the integral of it and plug in the numbers inbetween!
examples:
AREA:
y=-4x^2+41x+94 AND y=x-2 inbetween 7 and 1
so graph and then put the top equation over the bottom equation
the integral of (-4x^2+41x+94)-(x-2)dx
simplify: the integral of (-4x^2+40x+96) dx
((-4/3)(x)^3+20(x)^2+96(x)) of 1 and 7
f(7)=(3584/3) f(1)=(344/3)
ANSWER: 1080
VOLUME:
y=the sqr. root of (-2(x)^2-10(x)+48) inbetween 1 and -2
so graph and then put take the equation they give you into the integral and square it
PI times the integral of (-2(x)^2-10(x)+48) dx
PI [(-2/3)(x)^3-5(x)2+48(x)] of 1 and -2
f(1)=(127/3) f(-2)=(-332/3)
ANSWER: (153 PI)
hope this helps...[question is at the top]
~ELLIE~
Friday, December 4, 2009
ABOUT THE COMMENTS!
for some reason my computer didnt wanta let me get on the site..much less comment last night..so today i put them in...hope it's not too late!
Thursday, December 3, 2009
Ahhhh hmwk help!
Okay, I did part of last night's homework
Checked it today on Calcchat
Got it wrong!
I have no idea what I'm doing!!
Can someone PLEASE help with numbers 12-14??
Maybe 15 and 19 too, I haven't gotten there yet, I've spent forever on this dumb problems that refuse to solve -.-
Also, what in the world are the rectangles that calcchat uses??? I don't understand that and it makes me even more confused
I'm soooo lost! Can someone explain it?
Checked it today on Calcchat
Got it wrong!
I have no idea what I'm doing!!
Can someone PLEASE help with numbers 12-14??
Maybe 15 and 19 too, I haven't gotten there yet, I've spent forever on this dumb problems that refuse to solve -.-
Also, what in the world are the rectangles that calcchat uses??? I don't understand that and it makes me even more confused
I'm soooo lost! Can someone explain it?
HELP !
ok, i need some serious help on the homework... i do not understand how to find the bounds? If anyone can help me it would be a big help.. thanks
Integration and Volume
Here is a great site with visuals for the disk and washer method.
http://www.intmath.com/Applications-integration/4_Volume-solid-revolution.php
Also if anyone finds a copy of the program for mram, lram and rram let me know where.
Wednesday, December 2, 2009
Comments?
Ok, so the blog's being stupid and not letting me post comments?
I posted two comments on Ryan G.'s post, and none of them showed up
I posted two comments on Ryan G.'s post, and none of them showed up
Monday, November 30, 2009
week 14 and 15
this week was crunch time. we had the intense holiday packet to do and while doing it i was able to pick up on substitution.
substitution
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
Also something that i understand in calculus is intergration.
intergration is basically setting the derivative back to the origianl equation
you add one to the exponent and then multiply the term by the reciprocal of your exponent.
for instance - x^2 is 1/3x^3
one thing that i dont understand fully is mRam. I understand the process for the most part for lRAM and rRAM but mRAM i just dont get. I get lost and am not to certain of the steps.
substitution
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
Also something that i understand in calculus is intergration.
intergration is basically setting the derivative back to the origianl equation
you add one to the exponent and then multiply the term by the reciprocal of your exponent.
for instance - x^2 is 1/3x^3
one thing that i dont understand fully is mRam. I understand the process for the most part for lRAM and rRAM but mRAM i just dont get. I get lost and am not to certain of the steps.
14 andddd 15
man i just suck at internet stuff, i always forget.
holiday's are overr.. sad. but one step closer to graduationnn!
so for week 14 i'll just talk about riemans sum because that's really the only thing i remember before i left.
RRAM: right handed aprroximation
formula= delta x [ f(a + delta x) ... + f(b)]
EX: calculate the right tiemann sum for a(x)=3x on the interval [-2,3]
divided into 4 sub intervals. (sub intervals are just how many intervals you're setting up/how many numbers you're plugging in.
deltaX=5/4.. use (b-a)/# of subintervals to find this.
5/4[f(-3/4)+f(1/2)+f(7/4)+f(3)]
5/4[(9/4)+(3/2)+(21/4)+(9)= 81
LRAM: left handed approximation
formula= delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
MRAM: is to calculate the middleeee
formula= delta x [ f(mid) + f(mid) + .... ]
MMMM.. let's talk integrals!
yeahh, the fun S stuff.
it is basically the reverse action of a derivative.
.. there are two types: definite and indefinite.
Indefinite: the result of indefinite integration is an equation.
S(x^n)dx = [x^(n+1)]/[n+1] +C.
Definite: the result of definite integration is a number.
Definite - is just F(a) +F(b)= #
-take the intergral of a and b (x,y) and then plug it in to get an answer.
... mmm for #15 i wasnt there the week before thanksgiving so we're just going to pull some stuff from wayy back when, but it's okay because at the end i'll have tonnnss of questions. one thing i know i am is LOST. ohhh, and not to forget to put +C!
related rates:
1. Identify all of the variables and equations.
2. Identify what you want to find.
3. Sketch and label.
4. Write an equations involving your variables.
5. Take the derivative with in terms of time.
6. Substitute the derivative in and solve.
TANGENT LINES! MY FAVORITE
Steps:
Take f ‘(x).
Plug in x to find your slope: m.
Plug x into f(x) to find y (if not already given).
Using m and (x,y), plug it into slope-intercept form: (y-y1) = m(x-x1).
okay so for what i don't understand..
any new things with "graphs" i don't have a clue..
it'd be great if someone would like to go over what i missed those 4 days i was absent, you'd be a life saver and i'll bring you candy.
umm, and on the packet we had over the break.. i don't know what to do with 16-25..
anybody wanna help?
holiday's are overr.. sad. but one step closer to graduationnn!
so for week 14 i'll just talk about riemans sum because that's really the only thing i remember before i left.
RRAM: right handed aprroximation
formula= delta x [ f(a + delta x) ... + f(b)]
EX: calculate the right tiemann sum for a(x)=3x on the interval [-2,3]
divided into 4 sub intervals. (sub intervals are just how many intervals you're setting up/how many numbers you're plugging in.
deltaX=5/4.. use (b-a)/# of subintervals to find this.
5/4[f(-3/4)+f(1/2)+f(7/4)+f(3)]
5/4[(9/4)+(3/2)+(21/4)+(9)= 81
LRAM: left handed approximation
formula= delta x [f(a) + f( delta x +a) .... + f( delta x - b)]
MRAM: is to calculate the middleeee
formula= delta x [ f(mid) + f(mid) + .... ]
MMMM.. let's talk integrals!
yeahh, the fun S stuff.
it is basically the reverse action of a derivative.
.. there are two types: definite and indefinite.
Indefinite: the result of indefinite integration is an equation.
S(x^n)dx = [x^(n+1)]/[n+1] +C.
Definite: the result of definite integration is a number.
Definite - is just F(a) +F(b)= #
-take the intergral of a and b (x,y) and then plug it in to get an answer.
... mmm for #15 i wasnt there the week before thanksgiving so we're just going to pull some stuff from wayy back when, but it's okay because at the end i'll have tonnnss of questions. one thing i know i am is LOST. ohhh, and not to forget to put +C!
related rates:
1. Identify all of the variables and equations.
2. Identify what you want to find.
3. Sketch and label.
4. Write an equations involving your variables.
5. Take the derivative with in terms of time.
6. Substitute the derivative in and solve.
TANGENT LINES! MY FAVORITE
Steps:
Take f ‘(x).
Plug in x to find your slope: m.
Plug x into f(x) to find y (if not already given).
Using m and (x,y), plug it into slope-intercept form: (y-y1) = m(x-x1).
okay so for what i don't understand..
any new things with "graphs" i don't have a clue..
it'd be great if someone would like to go over what i missed those 4 days i was absent, you'd be a life saver and i'll bring you candy.
umm, and on the packet we had over the break.. i don't know what to do with 16-25..
anybody wanna help?
15th post
holidays!!!!!! are over and reality is unfortunaltly back so.....here we go again with the almighty calculus blog and yes i am doing on monday because i forgot....
LRAM is left hand approximation and the formula is:
delta 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.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e integration:
whatever is raised to the e power will be your u and du will be the derivative of u. For example:
e^2x-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
related rates:
The steps for related rates are….
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
limits:
Rules for Limits:…
1. if the degree of top equals the degree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
so I am still not understanding e integration and even though I have an example up im still having problem working them.
LRAM is left hand approximation and the formula is:
delta 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.
Substitution takes the place of the derivative rules for problems such as product rule and quotient rule. The steps to substitution are:
1. Find a derivative inside the interval
2. set u = the non-derivative
3. take the derivative of u
4. substitute back in
e integration:
whatever is raised to the e power will be your u and du will be the derivative of u. For example:
e^2x-1dx
u=2x-1 du=2
rewrite the function as:
1/2{ e^u du, therefore
1/2e^2x-1+C will be the final answer.
related rates:
The steps for related rates are….
1. Pick out all variables
2. Pick out all equations
3. Pick out what you are looking for
4. Sketch a graph and label
5. Create an equation with your variables
6. Take the derivative respecting time
7. Substitute back into the derivative
8. Solve
limits:
Rules for Limits:…
1. if the degree of top equals the degree of bottom, the answer is the top coefficient over bottom coefficient
2. if top degree is bigger than bottom degree, the answer is positive or negative infinity
2. if top degree is less than bottom degree, the answer is 0
so I am still not understanding e integration and even though I have an example up im still having problem working them.
Calculus Blog - Makeup
So the last week was amazing...I completely forgot about school, and it felt great :-)
Looks like in the process, I completely forgot about doing my blog as well :-)
Anyway, so I guess what I can explain is... e^x integration
For example:
integral of e^2x is just (1/2)e^2x+C
The derivative of e^x is just e^x times derivative of x.
So the integral is the opposite of its derivative times e^x. Kind of hard to explain...it just works....
Like for example...The integral of
(2x+4)e^(x^2+4x) is just e^(x^2+4x) + c because the derivative of u was already there. You only have to put the opposite whenever its not there...like the problem before.
Had it been 2e^2x then the integral would just be e^2x + c. You can check your answer by taking the derivative of that...which it works out.
The other thing I understand pretty well is ln integration...
1/x is ln|x|
So 2/x is 2ln|x|+C and 2/(1-x) is 2ln|1-x|+c
So anyway, ln integration is easy...
What else is there...
Today we learned how to finally do those ones with d/dx in front of the integral symbol...it turned out really easy actually...
You just cancel the integration and the d/dx and you plug in the top x or x^2 or whatever it is. So that's easy.
Overall the integration packet was not that bad...just really long. The rram, lram, mram, and trapezoidal ram took the longest because of crazy fractions and stuff though.
Anyway, that wraps up my post.
-John
Looks like in the process, I completely forgot about doing my blog as well :-)
Anyway, so I guess what I can explain is... e^x integration
For example:
integral of e^2x is just (1/2)e^2x+C
The derivative of e^x is just e^x times derivative of x.
So the integral is the opposite of its derivative times e^x. Kind of hard to explain...it just works....
Like for example...The integral of
(2x+4)e^(x^2+4x) is just e^(x^2+4x) + c because the derivative of u was already there. You only have to put the opposite whenever its not there...like the problem before.
Had it been 2e^2x then the integral would just be e^2x + c. You can check your answer by taking the derivative of that...which it works out.
The other thing I understand pretty well is ln integration...
1/x is ln|x|
So 2/x is 2ln|x|+C and 2/(1-x) is 2ln|1-x|+c
So anyway, ln integration is easy...
What else is there...
Today we learned how to finally do those ones with d/dx in front of the integral symbol...it turned out really easy actually...
You just cancel the integration and the d/dx and you plug in the top x or x^2 or whatever it is. So that's easy.
Overall the integration packet was not that bad...just really long. The rram, lram, mram, and trapezoidal ram took the longest because of crazy fractions and stuff though.
Anyway, that wraps up my post.
-John
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