My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree1Thanks
  • 1 Post By romsek
Reply
 
LinkBack Thread Tools Display Modes
October 10th, 2017, 11:53 AM   #1
Senior Member
 
Joined: Jan 2017
From: Toronto

Posts: 140
Thanks: 2

2D four leaves area problem

Find the area inside the four-leaf rose $\displaystyle r = \cos( 2 \theta ) $ and outside $r$ = 1/2.

Answer: $\displaystyle \sqrt{3} / 4 + \pi / 6 $

My solution:

$\displaystyle
\int_{0}^{ \pi /4 } \int_{1/2}^{ \cos {2 \theta} } 8 ~r~dr~d \theta = \pi /4
$

What did I do wrong?

Last edited by skipjack; October 10th, 2017 at 06:08 PM.
zollen is offline  
 
October 10th, 2017, 02:56 PM   #2
Senior Member
 
Joined: Jan 2017
From: Toronto

Posts: 140
Thanks: 2

I got it!!!

$\displaystyle
\int_{0}^{ \pi /6 } \int_{0}^{ \cos {2 \theta } } 8 ~r~dr~d \theta - \int_{0}^{ \pi / 6 } \int_{0}^{1/2} 8 ~r~dr~d \theta = \sqrt{3} / 4 + \pi / 6
$
zollen is offline  
October 10th, 2017, 03:08 PM   #3
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,488
Thanks: 749

romsek is online now  
October 10th, 2017, 10:55 PM   #4
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,488
Thanks: 749

Quote:
Originally Posted by zollen View Post
I got it!!!

$\displaystyle
\int_{0}^{ \pi /6 } \int_{0}^{ \cos {2 \theta } } 8 ~r~dr~d \theta - \int_{0}^{ \pi / 6 } \int_{0}^{1/2} 8 ~r~dr~d \theta = \sqrt{3} / 4 + \pi / 6
$
there is a clearer way of writing this

$\displaystyle 4 \int_{-\pi/6}^{\pi/6}\int_{1/2}^{\cos(2\theta)}~r~dr~d\theta$

The integral is one entire petal of the rose. There are 4 of them.
Thanks from zollen
romsek is online now  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
area, leaves, problem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Help with an area problem scsims Algebra 5 February 28th, 2017 06:56 AM
Area problem Imaxium Algebra 8 May 22nd, 2013 04:12 AM
Area problem taylor_1989_2012 Algebra 11 March 20th, 2013 12:53 AM
Area Problem julian21 Algebra 1 October 15th, 2010 11:16 PM





Copyright © 2017 My Math Forum. All rights reserved.