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how to calculate surface area bounded by $g(theta) = (cos(theta) , sin^3(theta))$

Calculate Area of SurfaceCompute the surface area of an oblate paraboloidHelp calculating the surface area given by the polar curve: $r=2(1-costheta)$Calculate surface area of a sphere using the surface integralCalculating surface areaFinding the surface area of $S=(rcostheta,rsintheta,3−r):0leq r leq 3, 0leq thetaleq2 pi $Surface area of a cone intersecting a horizontal cylindersurface area of sphereSurface area of $x^2 + y^2 = 4, quad 0 leq z leq 3$How to calculate the surface area of parametric surface?

0

$begingroup$

let $g(theta) = (cos(theta) , sin^3(theta))$ find the surface area bounded by this curve.
$ 0 leq theta leq 2pi $

this was an exam question from today i had VERY hard time calculating the integral and i failed to calculate . i want to be sure about the way i solved this question :

Let $ S(r,theta) = (rcos(theta) , rsin^3(theta))$ the surface we want to find its area.

$ 0 leq theta leq 2pi $

$ 0 leq r leq 1 $

$ ||S_rtimes S_theta|| = 2rsin^2(theta)cos^2(theta) + rsin^2(theta) $

so S = $ int_0^2piint_0^1 2rsin^2(theta)cos^2(theta) + rsin^2(theta) ~dr~dtheta$

and i failed to calculate this , the final answer is $frac3pi4$ according to the integral calculator.

is my solution alright ?

multivariable-calculus surfaces

share|cite|improve this question

edited 2 days ago

Mather

asked 2 days ago

Mather Mather

4108

$endgroup$

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  $begingroup$
  anyone i really need to know it is the only question that i was afraid of . and sadly they are gonna take 5-6 points for sure because i didn't calculate it
  $endgroup$
  – Mather
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  yes i got that and i said that $ sin^2(theta) = $1 - cos^2(theta )$ so plugging that in $sin^2(theta) ^ 2 $
  $endgroup$
  – Mather
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Anyway, this is a plane curve, so I wouldn't bother with the parametrization of the surface. I would just sketch the curve to make sure there are no surprise self-intersections, and then use one of the ways of getting the area surrounded by a close curve. Those are usually taught together with Green's theorem in the plane. Have you covered those? That way it becomes a routine trig integral.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  i though about green therom but i couldn't find " Nicee " Vector field that would get the surface area through green therom
  $endgroup$
  – Mather
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, your method is all right, and gives the correct answer in the end.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  

 | 
show 4 more comments

0

$begingroup$

let $g(theta) = (cos(theta) , sin^3(theta))$ find the surface area bounded by this curve.
$ 0 leq theta leq 2pi $

this was an exam question from today i had VERY hard time calculating the integral and i failed to calculate . i want to be sure about the way i solved this question :

Let $ S(r,theta) = (rcos(theta) , rsin^3(theta))$ the surface we want to find its area.

$ 0 leq theta leq 2pi $

$ 0 leq r leq 1 $

$ ||S_rtimes S_theta|| = 2rsin^2(theta)cos^2(theta) + rsin^2(theta) $

so S = $ int_0^2piint_0^1 2rsin^2(theta)cos^2(theta) + rsin^2(theta) ~dr~dtheta$

and i failed to calculate this , the final answer is $frac3pi4$ according to the integral calculator.

is my solution alright ?

multivariable-calculus surfaces

share|cite|improve this question

edited 2 days ago

Mather

asked 2 days ago

Mather Mather

4108

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  anyone i really need to know it is the only question that i was afraid of . and sadly they are gonna take 5-6 points for sure because i didn't calculate it
  $endgroup$
  – Mather
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  yes i got that and i said that $ sin^2(theta) = $1 - cos^2(theta )$ so plugging that in $sin^2(theta) ^ 2 $
  $endgroup$
  – Mather
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Anyway, this is a plane curve, so I wouldn't bother with the parametrization of the surface. I would just sketch the curve to make sure there are no surprise self-intersections, and then use one of the ways of getting the area surrounded by a close curve. Those are usually taught together with Green's theorem in the plane. Have you covered those? That way it becomes a routine trig integral.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  i though about green therom but i couldn't find " Nicee " Vector field that would get the surface area through green therom
  $endgroup$
  – Mather
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, your method is all right, and gives the correct answer in the end.
  $endgroup$
  – Jyrki Lahtonen
  2 days ago
  

  
  
  
  

 | 
show 4 more comments

$begingroup$

let $g(theta) = (cos(theta) , sin^3(theta))$ find the surface area bounded by this curve.
$ 0 leq theta leq 2pi $

this was an exam question from today i had VERY hard time calculating the integral and i failed to calculate . i want to be sure about the way i solved this question :

Let $ S(r,theta) = (rcos(theta) , rsin^3(theta))$ the surface we want to find its area.

$ 0 leq theta leq 2pi $

$ 0 leq r leq 1 $

$ ||S_rtimes S_theta|| = 2rsin^2(theta)cos^2(theta) + rsin^2(theta) $

so S = $ int_0^2piint_0^1 2rsin^2(theta)cos^2(theta) + rsin^2(theta) ~dr~dtheta$

and i failed to calculate this , the final answer is $frac3pi4$ according to the integral calculator.

is my solution alright ?

multivariable-calculus surfaces

share|cite|improve this question

edited 2 days ago

Mather

asked 2 days ago

Mather Mather

4108

$endgroup$

share|cite|improve this question

edited 2 days ago

Mather

asked 2 days ago

Mather Mather

4108

share|cite|improve this question

edited 2 days ago

Mather

asked 2 days ago

Mather Mather

4108

asked 2 days ago

Mather Mather

4108

asked 2 days ago

Mather Mather

4108