Volume and a ocomple and poly question (1 Viewer)

[GaganDeep]

The regionj bounded by the curce y=lnx and the line y=1 and the coordinate axes is rotated about the x axis.
Find volume by cyclingercial shells

I can';t do part d of this, the rest of it is alright




my thought
for that x^4 + Y^4 =1 not sure if right
op^4= (x^2 +y^2)^2=1+2x^2y^2
where 2>1+2X^2y^2
but that makes it op>2^(.25)?
not sure how the nsxt one

 

[pLuvia]

First one:
δV=2pixyδy
V=int.{0 to 1}2pi(e^y)ydy
Solve this and should get your answer

 

[zeek]

For the first part of the second image:

(x-y)²=>0
.: x²+y²-2xy=>0
.: x²+y²=>2xy
Let x=x² and y=y²
.: x⁴+y⁴=>2x²y²

I can't read the second part... it comes up blurred

 

[GaganDeep]

i can do that part lol
ii)if p is ony the curce X^4 +y^4 =1 show that OP<_2^(1/4) o is origin
second part express 1+x+x^2 +x^3 +x^4+ x^5 as a product of real factors
hence proff c+x+x^2/2 +x^3 /3 +...x^6 /6 =0 has no real roots if c>37/60

 

[KFunk]

x⁴ + y⁴ = 1 (1)


OP = (x² + y²)^1/2

OP⁴ = x⁴ + y⁴ + 2x²y²

OP⁴ = 1 + 2x²y² (2)


We know that x⁴ + y⁴ >= 2x²y² hence, using (2), it follows that :

x⁴ + y⁴ >= OP⁴ - 1

1 >= OP⁴ - 1 (using (1))

OP <= 2^1/4

 

[onebytwo]

that was a similar question (maybe the same) in sydney boys trial 2005

 

[NickP101]

Haha i still cant do question d on the first image lol its driving me nuts, i know ive done that question before!

 

[sasquatch]

Haha i still cant do question d on the first image lol its driving me nuts, i know ive done that question before!

FUCK me too man

i figured it out though..took me half an hour and now i see it :| fuck its so easy..


from c)

(1-w)(1-w⁴) = 2 - 2cos(2pi/5) = 2 - 2[1-2sin²(pi/5)]
= 2 - 2 + 4sin²(pi/5)
= 4sin²(pi/5)

Similarily,

(1-w²)(1-w³) = 2 - 2cos(4pi/5) = 2 - 2[1-2sin²(2pi/5)]
= 2 - 2 + 4sin²(2pi/5)
=4sin²(2pi/5)

From a),

(1-w)(1-w²)(1-w³)(1-w⁴) = 16sin²(pi/5)sin²(2pi/5) = 5
sin²(pi/5)sin²(2pi/5) = 5/16
sin(pi/5)sin(2pi/5) = root5 / 4

SEE fuck man, good revision though..cuz i always get screwed with stupid roots of unity...

 

[NickP101]

Thanks man!

 

[GaganDeep]

lol thanks, what about
part express 1+x+x^2 +x^3 +x^4+ x^5 as a product of real factors
hence proff c+x+x^2/2 +x^3 /3 +...x^6 /6 =0 has no real roots if c>37/60

 

[Riviet]

x⁵+x⁴+...+x+1=x³(x²+x+1) + x²+x+1
=(x³+1)(x²+x+1)
=(x+1)(x²-x+1)(x²+x+1)

Not fully sure about the next bit but if you integrate x⁵+x⁴+...+x+1=0 and sub in -1 as it was a root of the previous expression (see factorised form), you find that the constant (c) is -37/60.