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anonymous
 4 years ago
Calculus II  Volume of Solids  Setup Question
\[y = 4xx^2\]
\[y=2x\]
Let D be the region bound by the curves above. Setup an integral representing the volume of the solid obtained by revolving R about each of the following lines;
a) the xaxis
b) the line y=5
c) the line y = 1
d) the line x = 1
e) the line x = 3
I have worked all five using the "washer method" and would like to get some feedback on the answers;
a) \[\pi \int\limits_{1}^{2} (4x^2)^2  (2x)^2 \]
b) \[\pi \int\limits_{1}^{2} (2x)^2  (5(4x^2))^2 \]
c) \[\pi \int\limits_{1}^{2} (2+x)^2  (1+(4x^2))^2 \]
anonymous
 4 years ago
Calculus II  Volume of Solids  Setup Question \[y = 4xx^2\] \[y=2x\] Let D be the region bound by the curves above. Setup an integral representing the volume of the solid obtained by revolving R about each of the following lines; a) the xaxis b) the line y=5 c) the line y = 1 d) the line x = 1 e) the line x = 3 I have worked all five using the "washer method" and would like to get some feedback on the answers; a) \[\pi \int\limits_{1}^{2} (4x^2)^2  (2x)^2 \] b) \[\pi \int\limits_{1}^{2} (2x)^2  (5(4x^2))^2 \] c) \[\pi \int\limits_{1}^{2} (2+x)^2  (1+(4x^2))^2 \]

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0d) \[\pi \int\limits_{1}^{2} (2x)^2  (1+(4x^2))^2 \] e) \[\pi \int\limits_{1}^{2} (2x)^2  (3(4x^2))^2 \]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2im assuming its the area bound by the intersection of these functions

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2washer method you have to make sure your integrating and "adjusting" according to the axis of roataion as well

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2a function viewed from the x axis doesnt look the same as a function viewed from the y axis; it becomes its inverse function

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Yes, bound by the intersection. Shoot, adjusting. I also tried this for D (shell method); d) \[2\pi \int\limits_{1}^{2} (x+1)[(4x^2)(2x)]\]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So anytime I am rotating about the Yaxis I need to adjust my equations to be in terms of Y?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Using the washer method, that is.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2adjust your equations according to axis and method of ...sheel, disc, washer etc

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2dw:1327958145327:dw

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2R = f(x) r = g(x) pi (R^2  r^2) = washer area integrated with the x axis

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2washer is just a double disc

TuringTest
 4 years ago
Best ResponseYou've already chosen the best response.1shell about x integrate w/respect to x disk about x integrate w/respect to y and viceversa

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2and look into your function from the way the method is looking at it; if its a radius define it from the radial axis

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Ok, so if you rotate around y axis R = f(y) r = g(y)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2if its a heaight, define it from the base axis

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2inverse when you are looking at it from the other axis that the equation is defined for

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2y=x^2 looking at it from the yaxis is x=sqrt(y)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2dw:1327958415125:dw

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So I am assuming A is correct, but for B; \[\pi \int\limits_{1}^{2}(y2)^2  (5(\sqrt(4y)))^2\] ?

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2dont forget the dy if thats the case

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0But tacking a dy onto the end its correct? :)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2y=4x−x^2 y=2−x  2+5xx^2 = 0 x^2 5x +2 = 0 x = 5/2 + sqrt(17)/2 you sure your intercepts are right? Let D be the region bound by the curves above. Setup an integral representing the volume of the solid obtained by revolving R about each of the following lines; a) the xaxis

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0We're actually given a picture to go along with the assignment; dw:1327958708347:dw

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2that picture doesnt match your equations that you posted

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Ugh. I attached an X. y = 4 + x^2 Really sorry about that, but it shouldn't have effected much other then making everything wrong :)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2lol, its right everywhere except where its wrong :)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2better, 1,3 and 2,0 are going to be our "limits" or at least we can keep them in mind

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2you want washer rotate about the x axis for a \[\pi\int_{1}^{2}\ (4x^2)^2(2x)^2 dx\] do you know what happens when you get the top and bottom in the wrong place?

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2yep, so does position matter too much?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Nope! :) Of course, some of the grade will be setting the equation up properly, I'm sure. But I see your point!

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2good, cause all the texts I see waste so much time trying to tell you to set up top on top and bottom on bottom or else the sun will explode or some nonesense. the difference between a 2 and a 2 is just drop the negative :)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Hahaha, exploding sun.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0So A and B are looking good, except for dx and dy respectively? Then for C I need to change to F(y) and D E should look good?

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2b) the line y=5 this is the same as the xaxis; xaxis IS y=0 y=5 is just another parallel xaxis; what I do is subtract 5 from the eqs to set it back up to y=0

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2y = 4x^2 5 5  y = 1x^2 same eq, different position

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2y =2x 5 5  y = 3x

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Damn, that's a great idea. So that prevents you from having to subtract it from 5. If I had not done that it is correct, though?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Can you do that with any Y or X = ? question. Just get the equation to F(y) or F(x) and add the x = or y = ?

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2you swapped top and bottom on your B and forgot to 5 from one of the eqs

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2yes, but with left and right shifts you have to modify the (x) perse say x^2 is measuered from x=5 instead of x=0 to move x^2 to the left by 5 we do: (x+5)^2

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2it just about knowing how to move the graph around to a good vantage point

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Yeah, I am TERRIBLE at visualizing/setting up these graphs. Trying to get better, which is why I am on here constantly asking for setup help. Everytime I leave this website though I feel more and more confident.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2so B, washered about y=5; our integration hasnt shifted left and right so x to x is still good; 1 to 2 \[\pi \int_{1}^{2} (4x^25)^2(2x5)^2 dx\]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2c) the line y = 1 this is the same thing; but +1 to move things to the right spot

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2\pi \int_{1}^{2} (4x^2+1)^2(2x+1)^2 dx

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Now when it is X = 1 we will need to get it in terms of Y and then add one?

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2forgot me delimiters :)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2one thing at time ...

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Haha, k. Trying to apply that principle to the next questions.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2\[B=\pi \int_{1}^{2} (4x^2+1)^2(2x+1)^2 dx\]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2that was C lol, but its a fancy looking mistake so its ok

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Beautiful, got the same thing written on my notepad in front of me.

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2d) the line x = 1 our radiuseses are now look from the y axis out so we have to invert the equations to proper perspectives

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I've got to go afk for 15 minutes (moving from Library to Work) but I will be right back! If you are not here, thank you SO MUCH! You have given me a MUCH greater understanding of these concepts and I truly feel like i could tackle almost any washer problem with minimal difficulty!

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2y = 4  x^2 y4 =x^2 4y = x^2 sqrt(4y) = x y = 2x y2 = x 2y = x ok; and im sure they want inner outer set up "correctly" as well

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2but yes, to adjust the eqs for x=1; we +1 to our eqs now

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0And then make sure the inner and outer equations are right. When doing inner/outer with respect to Y I assume we have to regraph the new equations and evaluate which is top and bottom?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ahhh, now brb! Going to be late :)

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2left and right; but yes

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2the graphs dont change; we just see them from the y axis instead of the x axis

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2of cours ethe math doesnt care if you use and x or y; just as long as your consistent

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2for D about x=1 \[\pi\int_{y=0}^{y=3}(\sqrt{4y}+1)^2(2y+1)^2dy\]

amistre64
 4 years ago
Best ResponseYou've already chosen the best response.2my ride is ready to leave, so thanks you for the time and good luck :)
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