Question

find the area of the region enclosed by the rose r=6cos3t

Answer 1

dont you use integrals?

Answer 2

ohhh, Rose curve.. right ??
compare it with r =acos(Kt)


Area = (Pi * a2) / 2 .............................. If K is even,,

Area = (Pi * a2) / 4 .............................. if K is odd..

i guess.. :-p

Answer 3

@lgbasallote : what say ?? its been long...

Answer 4

i don't get it

Answer 5

integrate within the limits of t....

Answer 6

r=6cos3t comparing it with.. r =acos(Kt)

a = 6
k = 3..

K is odd number..
SO,

Area = (Pi * a2) / 4 = (Pi * 6 sq) / 4 = (3.14 * 36) / 4 = 28.26...

????????????????????

Answer 7

http://en.wikipedia.org/wiki/Rose_(mathematics)

Answer 8

oh.. m right :)..

Answer 9

you want solve it by integration?

Answer 10

yes

Answer 11

t is angle ?

Answer 12

yes

Answer 13

first:
graph the curve:
http://www.wolframalpha.com/input/?i=r%3D6cos3t+graph+rose

Answer 14

since the area enclosed by the rose has 6 equal parts, we'll find one and multiply it by 6.

Answer 15

using the formula for polar curve's area:
\[A=\int\limits_{b}^{a} \frac{ 1 }{ 2 }r^2 dt\]

Answer 16

so,
\[Area = 6 \int\limits_{0}^{\frac{ \pi }{ 3 }} \frac{ 1 }{ 2 }(6cost)^2 dt\]

Answer 17

\[= 108\int\limits_{0}^{\frac{ \pi }{ 3 }} \cos^2 3t dt\]

Answer 18

\[= 36 \int\limits_{0}^{\frac{ \pi }{ 3 } } \cos^23t d(3t)\]

Answer 19

\[=36[\frac{ 1 }{ 2 } (3t) + \frac{ 1 }{ 4 } \sin (6t)]_{0}^{\frac{ \pi }{ 3 }}\]

Answer 20

\[= 36[ (\frac{ \pi }{ 2 }+0)-(0+0)]\]

Answer 21

\[=18 \pi\]

Answer 22

and there you go, all done!

Answer 23

This stunning

Answer 24

You are wonderful , but there are something i want ask about can i ?

Answer 25

go on ahead!

Answer 26

first how do you know the shape of the graph ?

Answer 27

you dont know any mathematical way to know it ?!

Answer 28

the question said rose, so it's a flower like curve, for how many petals you know it by r=a cos (nt) if n=3 then 3 petals

Answer 29

if n=2 then 4 petals, n=3 3 petals, n=4 8 petals, n=5 5 petals, n=6 12 petals, n=7 7 petals and so on

Answer 30

this sequence works only for a rose, for other polar curves, refer to your book, or other internet materials.

Answer 31

thats perfect , Iam sorry ,I studied integration past year alone and did not study these rules

Answer 32

nah, this isn't within the integration department, it's with the parametric and polar curves part, i think you learn this after integration.