1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

2 question in integrals

  1. Jan 23, 2008 #1
  2. jcsd
  3. Jan 23, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    For the first problem, find the length of the "astroid", x2/3+ y2/3[/sup]= a2/3, you say you wanted to put it in "polar form" and then give x= a cos(([itex]\alpha[/itex]), y= a sin([itex]\alpha[/itex]). That is NOT "polar form. Polar form would be x= r cos([itex]\alpha[/itex]), y= r sin([itex]\alpha[/itex]), where both r and [itex]\alpha[/itex] are independent variables. There is no reason to thing that x= a cos(([itex]\alpha[/itex]), y= a sin([itex]\alpha[/itex]) satisfy the equation of the curve.

    As for the "method" your text uses, you say they let x= a sin3(t), y= a cos3(t). Okay, then x2/3+ y2/3= a2/3sin2(t)+ a2/3cos2(t)= a2/3 so those are parametric equations for the curve. Also, then dx= 3a cos(t)sin(2(t)dt and dy= -3a sin(t)cos2(t) dt so that ds= [itex]\sqrt{(dx/dt)^2+ (dy/dt)^2}= \sqrt{9a^2 cos^2(t)sin^3(t)+ 9a^2 sin^2(t) cos^4(t)}dt[/itex].

    As for the second one, y2= x2(a2- x2, i don't see why you again have "x= a cos[itex]\alpha[/itex], y= a sin[itex]\alpha[/itex]". Polar coordinates are x= r cos[itex]\theta[/itex], y= r sin[\theta]. (of course, it doesn't matter whether you use [itex]\alpha[/itex] or [itex]\theta[/itex]. [itex]\theta[/itex] is the standard notation.)

    Then [itex]y^2= r^2 sin^2(\theta)[/itex] and [itex]x^2= r^2 cos^2(\theta)[/itex] so your equation becomes [itex]r^2 sin^2(\theta)= r^2(a^2- r^2 cos^2(\theta)[/itex]. We can cancel the two [itex]r^2[/itex] terms and then we have [itex]sin^2(\theta)= a^2- r^2 cos^2(\theta)[/itex] so [itex]r^2 cos^2(\theta)= a^2- sin^2(\theta)[/itex], [itex]r^2= a^2 sec^2(\theta)- tan^2(\theta)[/itex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: 2 question in integrals
  1. 2 integral questions (Replies: 4)

  2. Integral 2 (Replies: 14)

Loading...