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LogicalApple
Group Title
Ok the attached image is from a calculus textbook (http://www.stewartcalculus.com/media/11_home.php).
The question is simple enough  it says that this figure contains all points inside the square that are closer to the center than to the sides of the square. Find the area of this region.
Anyone have an idea on how to get started?
The image will be uploaded shortly.
 one year ago
 one year ago
LogicalApple Group Title
Ok the attached image is from a calculus textbook (http://www.stewartcalculus.com/media/11_home.php). The question is simple enough  it says that this figure contains all points inside the square that are closer to the center than to the sides of the square. Find the area of this region. Anyone have an idea on how to get started? The image will be uploaded shortly.
 one year ago
 one year ago

This Question is Closed

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
I think, you'll need to do 2 integrals, and then using symmetry you should be able to apply these to the rest of the diagram. However, I'm not sure of the best way to set up the integrals. dw:1356844485378:dw This picture very roughly imitates your shape. The areas I've shaded need to be integrated, each one separately. The area on the left you can find by looking at the center of the square and the left side of the box. The area in the middle can be found by looking at the center of the square and the upper part of the square. You just need to find the equation of the line that is equidistant from the corresponding sides of the square and the center of the square. If I remember correctly, these should take the form of hyperbolas.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Alright, I've got to go. Hopefully this was enough to get you started. I'm pretty sure that the lines you're looking for will be hyperbolic. Then you just have a couple definite integrals to solve.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Thank you for your time!
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I will work on this right away
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
It also might be interesting to note that a parabola is defined as a curve where all points are equidistant to a focus (point) and a directrix (line). so you can tell that these lines are parabolas.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Hm. I was thinking parabolas, too. I think that might be just what I am looking for.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
From my calculus book: An equation of a (horizontal) parabola with focus \((0, p)\) and directrix \(y = p\) is: \(x^2 = 4py\).
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Let the very bottom line be the directrix and let the very center be the focus...
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
The center and directrix are 1 unit apart, so \[ p(p) = 1 \\ 2p = 1 \\ p = 1/2 \] Obviously \(p\). can't be negative
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ x^2 = 4(1/2)y = 2y \implies y = x^2/2 \]So this gives us the equation of the bottom line when the xaxis is the very center, and the y axis is halfway between the center (focus) and the bottom (directrix). @LogicalApple I think you can do the rest from here.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Thank you so much!
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
No problem, it was a bit of a learning experience for me as well.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Would you agree with the points I drew in and labeled ?
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Actually, let me think about it for a moment...
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
I'm not sure of the logic used to get those points so I can't really confirm or deny... I am thinking perhaps you could find where the two parabolas would intercept.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I based my calculations on these points. I had enough points to generate parabolas. One parabola was x = (2  sqrt 2)y^2  1 This would he the parabola on the far left side. Since I'm only interested in a portion of it (namely from when x = 1 to when x = sqrt (2) / 2) I could integrate y = sqrt( (1 + x) / (2  sqrt (2)) from 1 to sqrt (2)/2 Multiply this result by 4 and that would take care of the 4 triangular regions (I only drew two of them in the figure). For the parabola hanging overhead from x = sqrt(2)/2 to sqrt(2)/2 I calculated y = (sqrt 2  2)x^2 + 1. Integrate this from sqrt(2)/2 to sqrt(2)/2 and multiply the result by 2. This takes care of the remainder of the region I think. My end result was ( 8 sqrt(2)  2)/3 But I have no way of verifying this
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
be the*
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
The boundary of this figure is equally as far from the center as it is from the perimeter of the square. So along the horizontal axis, the point (1, 0) is as far from (0, 0) as it is from (2, 0). I suppose I could have used (0, 1) as another point.... Instead I determined the point of the corner of the boundary as half of the diagonal of a square of hypotenuse 4 (2^2 + 2^2).
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Wait now that I think about it..
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Those numbers should be cut in half
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356849810908:dw I know it's not a circle but think of it this way.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
You're absolutely right. Back to the drawing board for me.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356849937491:dw
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356850006505:dw
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
In this case, it was just finding the right algebra.... lol
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ \sqrt{x^2 + x^2} + x = \sqrt{1^2+1^2} = \sqrt{2} \]
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Right, sqrt(2) . Hm.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Yeah, well, just solve for \(x\)...
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ \sqrt{x^2+x^2}+x = \sqrt{2x^2}+x = \sqrt{2} \cdot x+x \]Thus \[ (1+\sqrt{2})x = \sqrt{2} \]@LogicalApple Make sense... convinced? Don't assume I'm a master, either. Look at what I did to check for mistakes.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
@LogicalApple Hmm?
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I'm still trying to figure out why x + y = sqrt(1^2 +1^2) instead of x + y = sqrt(2^2 + 2^2)?
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
More stuff:\[ (\sqrt{2}+1)x = \sqrt{2} \\ (\sqrt{2}1)(\sqrt{2}+1)x=(\sqrt{2}1)\sqrt{2} \\ (21)x = 2\sqrt{2} \\ x = 2\sqrt{2} \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Because the total length is 2, but we're only doing halfway... which is 1.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Yeah scratch that
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I am in agreement that x = 2  sqrt 2
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I will modify my equations
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Okay, so what we can do if first just find the area of the inner square. Then we can find the area between one of the parabola and the inner square... and by symmetry just multiply that by 4.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356850703108:dw
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
4x as in 4 times
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I like that idea a lot actually
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
The area of the square is easy. We just found half the length of its side. So just double that and square it.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
4(2  sqrt 2)^2
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
We also already found the equation of the parabola, and we know its limits.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
@LogicalApple Got an idea for finding the area under that parabola?
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I'm still uncertain of the equation of the parabola. I calculate something like: y = (sqrt 2  1) / (4 sqrt 2  6) * x^2 + 1
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
I don't know how you're getting that...
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
This parabola contains the points (2 + sqrt 2, 2  sqrt 2), (0, 1), and (2  sqrt 2, 2  sqrt 2) The top two corner points of the square and the vertex of the parabola
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I use a system of 3 equations. Where the general form is f(x) = ax^2 + bx + c I let x = 2 + sqrt 2, f(x) = 2  sqrt 2 x = 0, f(x) = 1 x = 2  sqrt 2, f(x) = 2  sqrt (2)
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
We could just ignore that it a parabola altogether and just try using the distance formula.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Distance of some point \((x, y)\) to the line \(y=1\) \[ \sqrt{(xx)^2(y1)^2} \]Distance of some point \((x, y)\) to the origin \[ \sqrt{x^2+y^2} \]Set them equal \[ \sqrt{(xx)^2(y1)^2} = \sqrt{x^2+y^2} \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Should be a plus in there.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Now I think I'm getting a rational answer of 8/3 for the area of the region.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ \sqrt{(y1)^2} = \sqrt{x^2 + y^2} \\ (y1)^2 = x^2 + y^2 \\ y^2  2y + 1 = x^2 + y^2 \\ 2y + 1 = x^2 \\ y = \frac{x^2}{2} +\frac{1}{2} \]We got a different equation because we had a different origin.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
But notice how our slope is pretty much the same.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Just reflected because this time we took the concave down parabola at top.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I'm following.. and I'm curious. I am going to review this entire post when it's all said and done.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
To find the area under parabola \(y = x^2/2 + 1/2\) and above the line \(y = 2  \sqrt{2} \), we take the integral: \[ \int_{1(2\sqrt{2})}^{2\sqrt{2}} \left[ \frac{x^2 + 1}{2}  (2\sqrt{2}) \right]dx \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ \int_{(2\sqrt{2})}^{2\sqrt{2}} \left[ \frac{x^2 + 1}{2}  (2\sqrt{2}) \right]dx \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Sorry, been having quite a few typos today.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Don't sweat it
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
That's a pretty easy integral... but it just has a lot of sqrts and stuff that I am a bit reluctant to do it.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Right but I can just use symmetry again. Plus there is wolfram alpha
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
No it's an extremely easy integral it's just:\[ \int_{(2\sqrt{2})}^{2\sqrt{2}} \left[ \frac{x^2 + 1}{2}  (2\sqrt{2}) \right]dx \\ = \left[ \frac{x^3}{3\times 2} + \frac{x}{2}  (2\sqrt{2})x \right] _{(2\sqrt{2})}^{2\sqrt{2}} \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
plugging that pellet in that is gonna be a pain in the retrice..
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Anyway, I know you can do the rest from here. Just look things over a couple times if you don't get anything I said. There maybe some typos but the general methodology is correct as far as I can see.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
\[\int\limits_{(2  \sqrt{2)}}^{2  \sqrt{2}} \frac{ \sqrt{2}  1 }{ 4\sqrt{2}  3} x^{2}dx \int\limits_{(2  \sqrt{2)}}^{2  \sqrt{2}} (2  \sqrt{2}) dx = 4\sqrt{2}  16\] The above integral is based on the attached picture (below) It determines the area of one of the parabolic regions bounded by the inner square. Multiply this result by 4 and add the area of the square, we obtain 8/3 as the area of the region.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
should be a dx in that second integral (sorry am still getting used to the equation creator)
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Oh wait there is already one there. Nvm. Anyway, thanks a lot to @wio for your patience. This was an unusually challenging problem but you found a way to simplify the heck out of it! And thanks to KingGeorge for visualizing it and getting me on the right track. OpenStudy is pretty cool..
 one year ago

dumbcow Group TitleBest ResponseYou've already chosen the best response.0
after looking this over...i agree with the methods except i believe the intersection points are incorrect the limits should be +(sqrt2 1) Not +(2sqrt2) when \[\frac{1}{2}x^{2}+\frac{1}{2} = x\] \[x = \sqrt{2}  1\]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
That looks right... makes me wonder where I went wrong.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I think your geometry and algebra are correct in concluding the corner points of the region do lay at + (2  sqrt 2). I will double check.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
@LogicalApple They are somehow wrong. @dumbcow 's method makes more sense.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
But didn't you calculate the distance from a point to the line y = 1 instead of to the line y = 2?
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Of course you did. Hm.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Let me see what I obtain next.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Wow it's really bothering me how I messed up. I can't figure out why my trig method was wrong.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
@dumbcow 's method makes sense... you want the point where our parabola intersects y = x... But my method seemed correct as well... gotta figure out the fallacy I made.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
Ooooh! I figured out my error!
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356855949526:dw We found \(x\)! We wanted \(z\) in the diagram!
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
dw:1356856038380:dw
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
\[ 2z^2 = x^2 \\ \sqrt{2}\cdot z= x \\ z = x/\sqrt{2} = \frac{2\sqrt{2}}{\sqrt{2}} = \frac{2}{\sqrt{2}}  1 = \sqrt{2}1 \]Lol... really worry about that @LogicalApple Thank @dumbcow for spotting it.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Modifying .. 1 sec
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I did worry about it!
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
That looks correct.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I will integrate now.. and I have a feeling it will be irrational this time ;(
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I now see why there was a discrepancy with my earlier parabolic equation.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
I knew from the very start there was very little chance of a rational result.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
My image is still wrong. One more modification ..
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
That's pretty much it. It's an easy integral because it's just a 2 degree polynomial.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
Final answer is ... \[\frac{ 4 }{ 3 }(4\sqrt{2}  5) \]
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
For just the parabola?
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
That is 4 times the parabola plus the area of the square.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
It lists your answer as an alternate form.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
\[\int\limits_{(\sqrt{2}  1)}^{\sqrt{2} 1} (\frac{ 1  x ^{2} }{ 2}) dx  \int\limits_{(\sqrt{2}  1)}^{\sqrt{2} 1} (\sqrt{2}  1)dx = \frac{ 10 \sqrt{2} 14 }{ 3 }\] Oh yeah, Wolfram is great! The above should be the area of just one parabolic region. Multiplying this by 4 and then adding the area of the square yields the previous result.
 one year ago

wio Group TitleBest ResponseYou've already chosen the best response.1
http://www.wolframalpha.com/input/?i=%5Cint+from+%28sqrt%282%29+1%29+to+%28sqrt%282%291%29+++%5B+%281x%5E2%29%2F2++%28sqrt%282%29++1%29%5Ddx Wolfram agreeing with that too. Good algebra.
 one year ago

LogicalApple Group TitleBest ResponseYou've already chosen the best response.0
I feel like something was accomplished tonight. Then again, I wonder what the one guy felt like spending 7 years solving Fermat's last theorem....
 one year ago
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