How to calculate the surface area of parametric surface?











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Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2leq 1$.



My homework is forcing me to use the parameterization



$$textbf{r}_1(s,t)= <scos(t), ssin(t), 3s^2sin(t)cos(t)>$$



I am having a difficult time visualizing this parameterization, and I do not have any graphing software to graph the surface, but I want to make sure I understand this concept.



This is quite obvious, but I want to be sure; using the above parameterization, I am not parameterizing the entire surface, right? If I wanted to, I assume the parameterization would be $$textbf{r}_2(s,t) = <s,t,3st>$$



Instead, is $textbf{r}_1$ just the parameterization adjusted for the region - the region being the cylinder $x^2+y^2leq 1$? That is, are we just making a revolution around $z=3xy$?



Any insight would be helpful.










share|cite|improve this question


























    up vote
    3
    down vote

    favorite












    Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2leq 1$.



    My homework is forcing me to use the parameterization



    $$textbf{r}_1(s,t)= <scos(t), ssin(t), 3s^2sin(t)cos(t)>$$



    I am having a difficult time visualizing this parameterization, and I do not have any graphing software to graph the surface, but I want to make sure I understand this concept.



    This is quite obvious, but I want to be sure; using the above parameterization, I am not parameterizing the entire surface, right? If I wanted to, I assume the parameterization would be $$textbf{r}_2(s,t) = <s,t,3st>$$



    Instead, is $textbf{r}_1$ just the parameterization adjusted for the region - the region being the cylinder $x^2+y^2leq 1$? That is, are we just making a revolution around $z=3xy$?



    Any insight would be helpful.










    share|cite|improve this question
























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2leq 1$.



      My homework is forcing me to use the parameterization



      $$textbf{r}_1(s,t)= <scos(t), ssin(t), 3s^2sin(t)cos(t)>$$



      I am having a difficult time visualizing this parameterization, and I do not have any graphing software to graph the surface, but I want to make sure I understand this concept.



      This is quite obvious, but I want to be sure; using the above parameterization, I am not parameterizing the entire surface, right? If I wanted to, I assume the parameterization would be $$textbf{r}_2(s,t) = <s,t,3st>$$



      Instead, is $textbf{r}_1$ just the parameterization adjusted for the region - the region being the cylinder $x^2+y^2leq 1$? That is, are we just making a revolution around $z=3xy$?



      Any insight would be helpful.










      share|cite|improve this question













      Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2leq 1$.



      My homework is forcing me to use the parameterization



      $$textbf{r}_1(s,t)= <scos(t), ssin(t), 3s^2sin(t)cos(t)>$$



      I am having a difficult time visualizing this parameterization, and I do not have any graphing software to graph the surface, but I want to make sure I understand this concept.



      This is quite obvious, but I want to be sure; using the above parameterization, I am not parameterizing the entire surface, right? If I wanted to, I assume the parameterization would be $$textbf{r}_2(s,t) = <s,t,3st>$$



      Instead, is $textbf{r}_1$ just the parameterization adjusted for the region - the region being the cylinder $x^2+y^2leq 1$? That is, are we just making a revolution around $z=3xy$?



      Any insight would be helpful.







      multivariable-calculus vectors surfaces parametrization






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      share|cite|improve this question











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      asked Nov 19 at 10:34









      Art

      3127




      3127






















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          You can see this exercise as a surface integral or you can also see it as a double integral of a function $f(x,y)$ over the unit disc.





          $mathbf r_1$ is the cylindrical parameterization, which, generally, is
          $$begin{cases}x=scos t\y=ssin t\z=f(x,y)=f(scos t,ssin t)end{cases}$$



          In your case $z=f(x,y)=3xy$. In $mathbf r_1$, the cylinder becomes $sle1$ and you have no limitations on $t$. You have to evaluate the integral
          $$begin{align}int_0^1stimes 3s^2mathrm dsint_0^{2pi}sin(t)cos(t)mathrm dt&=0end{align}$$



          I got to this integral by the following way:



          $$iint f(x,y)mathop{mathrm dx}mathop{mathrm dy}=iint 3xymathop{mathrm dx}mathop{mathrm dy}$$
          Using polar coordinates $(x=scos t, y=ssin t)$ and adding the Jacobian:
          $$iint3s^3sin tcos tmathop{mathrm ds}mathop{mathrm dt}$$
          Because we are in the unit disc, $0<sle1, quad0<t<2pi$
          $$int_0^13s^3left(int_0^{2pi}sin tcos tmathop{mathrm dt}right)mathop{mathrm ds}$$






          share|cite|improve this answer























          • The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
            – Christian Blatter
            Nov 19 at 11:57










          • I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
            – Lorenzo B.
            Nov 19 at 12:21












          • I understand your answer (+1). Why mine is wrong?
            – Lorenzo B.
            Nov 19 at 14:36










          • What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
            – Christian Blatter
            Nov 19 at 15:00




















          up vote
          4
          down vote













          Since in the end you have to integrate over the unit disc in the $(x,y)$-plane your source proposes to use polar coordinates instead of $x$ and $y$ as parameters. Then $x=scos t$, $y=ssin t$. In this way the idea $z=3xy$ $>(x^2+y^2leq1)$ translates into
          $${bf r}(s,t)=(scos t,ssin t,3s^2cos tsin t)qquad(0leq sleq 1, 0leq tleq2pi) .$$
          In order to find the area of this floppy disc $F$ we have to compute
          $${bf r}_s=(cos t,sin t, 6scos tsin t),quad {bf r}_t=bigl(-ssin t,scos t ,3s^2cos(2t)bigr)$$
          and then
          $${bf r}_stimes{bf r}_t=(ldots,ldots,ldots) .$$
          The area is then finally given as
          $${rm area}(F)=int_0^1int_0^{2pi}bigl|{bf r}_stimes{bf r}_tbigr|>dt>ds .$$
          The resulting integral will be simpler than dreaded.






          share|cite|improve this answer























          • Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
            – Lorenzo B.
            Nov 22 at 18:44













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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          You can see this exercise as a surface integral or you can also see it as a double integral of a function $f(x,y)$ over the unit disc.





          $mathbf r_1$ is the cylindrical parameterization, which, generally, is
          $$begin{cases}x=scos t\y=ssin t\z=f(x,y)=f(scos t,ssin t)end{cases}$$



          In your case $z=f(x,y)=3xy$. In $mathbf r_1$, the cylinder becomes $sle1$ and you have no limitations on $t$. You have to evaluate the integral
          $$begin{align}int_0^1stimes 3s^2mathrm dsint_0^{2pi}sin(t)cos(t)mathrm dt&=0end{align}$$



          I got to this integral by the following way:



          $$iint f(x,y)mathop{mathrm dx}mathop{mathrm dy}=iint 3xymathop{mathrm dx}mathop{mathrm dy}$$
          Using polar coordinates $(x=scos t, y=ssin t)$ and adding the Jacobian:
          $$iint3s^3sin tcos tmathop{mathrm ds}mathop{mathrm dt}$$
          Because we are in the unit disc, $0<sle1, quad0<t<2pi$
          $$int_0^13s^3left(int_0^{2pi}sin tcos tmathop{mathrm dt}right)mathop{mathrm ds}$$






          share|cite|improve this answer























          • The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
            – Christian Blatter
            Nov 19 at 11:57










          • I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
            – Lorenzo B.
            Nov 19 at 12:21












          • I understand your answer (+1). Why mine is wrong?
            – Lorenzo B.
            Nov 19 at 14:36










          • What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
            – Christian Blatter
            Nov 19 at 15:00

















          up vote
          1
          down vote



          accepted










          You can see this exercise as a surface integral or you can also see it as a double integral of a function $f(x,y)$ over the unit disc.





          $mathbf r_1$ is the cylindrical parameterization, which, generally, is
          $$begin{cases}x=scos t\y=ssin t\z=f(x,y)=f(scos t,ssin t)end{cases}$$



          In your case $z=f(x,y)=3xy$. In $mathbf r_1$, the cylinder becomes $sle1$ and you have no limitations on $t$. You have to evaluate the integral
          $$begin{align}int_0^1stimes 3s^2mathrm dsint_0^{2pi}sin(t)cos(t)mathrm dt&=0end{align}$$



          I got to this integral by the following way:



          $$iint f(x,y)mathop{mathrm dx}mathop{mathrm dy}=iint 3xymathop{mathrm dx}mathop{mathrm dy}$$
          Using polar coordinates $(x=scos t, y=ssin t)$ and adding the Jacobian:
          $$iint3s^3sin tcos tmathop{mathrm ds}mathop{mathrm dt}$$
          Because we are in the unit disc, $0<sle1, quad0<t<2pi$
          $$int_0^13s^3left(int_0^{2pi}sin tcos tmathop{mathrm dt}right)mathop{mathrm ds}$$






          share|cite|improve this answer























          • The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
            – Christian Blatter
            Nov 19 at 11:57










          • I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
            – Lorenzo B.
            Nov 19 at 12:21












          • I understand your answer (+1). Why mine is wrong?
            – Lorenzo B.
            Nov 19 at 14:36










          • What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
            – Christian Blatter
            Nov 19 at 15:00















          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          You can see this exercise as a surface integral or you can also see it as a double integral of a function $f(x,y)$ over the unit disc.





          $mathbf r_1$ is the cylindrical parameterization, which, generally, is
          $$begin{cases}x=scos t\y=ssin t\z=f(x,y)=f(scos t,ssin t)end{cases}$$



          In your case $z=f(x,y)=3xy$. In $mathbf r_1$, the cylinder becomes $sle1$ and you have no limitations on $t$. You have to evaluate the integral
          $$begin{align}int_0^1stimes 3s^2mathrm dsint_0^{2pi}sin(t)cos(t)mathrm dt&=0end{align}$$



          I got to this integral by the following way:



          $$iint f(x,y)mathop{mathrm dx}mathop{mathrm dy}=iint 3xymathop{mathrm dx}mathop{mathrm dy}$$
          Using polar coordinates $(x=scos t, y=ssin t)$ and adding the Jacobian:
          $$iint3s^3sin tcos tmathop{mathrm ds}mathop{mathrm dt}$$
          Because we are in the unit disc, $0<sle1, quad0<t<2pi$
          $$int_0^13s^3left(int_0^{2pi}sin tcos tmathop{mathrm dt}right)mathop{mathrm ds}$$






          share|cite|improve this answer














          You can see this exercise as a surface integral or you can also see it as a double integral of a function $f(x,y)$ over the unit disc.





          $mathbf r_1$ is the cylindrical parameterization, which, generally, is
          $$begin{cases}x=scos t\y=ssin t\z=f(x,y)=f(scos t,ssin t)end{cases}$$



          In your case $z=f(x,y)=3xy$. In $mathbf r_1$, the cylinder becomes $sle1$ and you have no limitations on $t$. You have to evaluate the integral
          $$begin{align}int_0^1stimes 3s^2mathrm dsint_0^{2pi}sin(t)cos(t)mathrm dt&=0end{align}$$



          I got to this integral by the following way:



          $$iint f(x,y)mathop{mathrm dx}mathop{mathrm dy}=iint 3xymathop{mathrm dx}mathop{mathrm dy}$$
          Using polar coordinates $(x=scos t, y=ssin t)$ and adding the Jacobian:
          $$iint3s^3sin tcos tmathop{mathrm ds}mathop{mathrm dt}$$
          Because we are in the unit disc, $0<sle1, quad0<t<2pi$
          $$int_0^13s^3left(int_0^{2pi}sin tcos tmathop{mathrm dt}right)mathop{mathrm ds}$$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 19 at 15:24

























          answered Nov 19 at 10:57









          Lorenzo B.

          1,7472519




          1,7472519












          • The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
            – Christian Blatter
            Nov 19 at 11:57










          • I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
            – Lorenzo B.
            Nov 19 at 12:21












          • I understand your answer (+1). Why mine is wrong?
            – Lorenzo B.
            Nov 19 at 14:36










          • What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
            – Christian Blatter
            Nov 19 at 15:00




















          • The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
            – Christian Blatter
            Nov 19 at 11:57










          • I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
            – Lorenzo B.
            Nov 19 at 12:21












          • I understand your answer (+1). Why mine is wrong?
            – Lorenzo B.
            Nov 19 at 14:36










          • What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
            – Christian Blatter
            Nov 19 at 15:00


















          The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
          – Christian Blatter
          Nov 19 at 11:57




          The three upvotes and acceptance notwithstanding this answer is seriously wrong. Why should the area in question be $=0>$? We have a real "floppy disc" here!
          – Christian Blatter
          Nov 19 at 11:57












          I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
          – Lorenzo B.
          Nov 19 at 12:21






          I think the integrand is right, the limits of integration could be wrong. I tried to evaluate the integral in cartesian coordinates and I got $0$ again: $oint_{x^2+y^2le1}3xymathop{mathrm dxmathrm dy}=3int_{-1}^1xmathop{mathrm dx} int_{-sqrt{1-x^2}}^{sqrt{1-x^2}}ymathop{mathrm dy}=0$. I honestly do not see where the mistake is
          – Lorenzo B.
          Nov 19 at 12:21














          I understand your answer (+1). Why mine is wrong?
          – Lorenzo B.
          Nov 19 at 14:36




          I understand your answer (+1). Why mine is wrong?
          – Lorenzo B.
          Nov 19 at 14:36












          What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
          – Christian Blatter
          Nov 19 at 15:00






          What made you arrive at the integrand $3s^3cos tsin t>$? – That's my last word on this matter.
          – Christian Blatter
          Nov 19 at 15:00












          up vote
          4
          down vote













          Since in the end you have to integrate over the unit disc in the $(x,y)$-plane your source proposes to use polar coordinates instead of $x$ and $y$ as parameters. Then $x=scos t$, $y=ssin t$. In this way the idea $z=3xy$ $>(x^2+y^2leq1)$ translates into
          $${bf r}(s,t)=(scos t,ssin t,3s^2cos tsin t)qquad(0leq sleq 1, 0leq tleq2pi) .$$
          In order to find the area of this floppy disc $F$ we have to compute
          $${bf r}_s=(cos t,sin t, 6scos tsin t),quad {bf r}_t=bigl(-ssin t,scos t ,3s^2cos(2t)bigr)$$
          and then
          $${bf r}_stimes{bf r}_t=(ldots,ldots,ldots) .$$
          The area is then finally given as
          $${rm area}(F)=int_0^1int_0^{2pi}bigl|{bf r}_stimes{bf r}_tbigr|>dt>ds .$$
          The resulting integral will be simpler than dreaded.






          share|cite|improve this answer























          • Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
            – Lorenzo B.
            Nov 22 at 18:44

















          up vote
          4
          down vote













          Since in the end you have to integrate over the unit disc in the $(x,y)$-plane your source proposes to use polar coordinates instead of $x$ and $y$ as parameters. Then $x=scos t$, $y=ssin t$. In this way the idea $z=3xy$ $>(x^2+y^2leq1)$ translates into
          $${bf r}(s,t)=(scos t,ssin t,3s^2cos tsin t)qquad(0leq sleq 1, 0leq tleq2pi) .$$
          In order to find the area of this floppy disc $F$ we have to compute
          $${bf r}_s=(cos t,sin t, 6scos tsin t),quad {bf r}_t=bigl(-ssin t,scos t ,3s^2cos(2t)bigr)$$
          and then
          $${bf r}_stimes{bf r}_t=(ldots,ldots,ldots) .$$
          The area is then finally given as
          $${rm area}(F)=int_0^1int_0^{2pi}bigl|{bf r}_stimes{bf r}_tbigr|>dt>ds .$$
          The resulting integral will be simpler than dreaded.






          share|cite|improve this answer























          • Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
            – Lorenzo B.
            Nov 22 at 18:44















          up vote
          4
          down vote










          up vote
          4
          down vote









          Since in the end you have to integrate over the unit disc in the $(x,y)$-plane your source proposes to use polar coordinates instead of $x$ and $y$ as parameters. Then $x=scos t$, $y=ssin t$. In this way the idea $z=3xy$ $>(x^2+y^2leq1)$ translates into
          $${bf r}(s,t)=(scos t,ssin t,3s^2cos tsin t)qquad(0leq sleq 1, 0leq tleq2pi) .$$
          In order to find the area of this floppy disc $F$ we have to compute
          $${bf r}_s=(cos t,sin t, 6scos tsin t),quad {bf r}_t=bigl(-ssin t,scos t ,3s^2cos(2t)bigr)$$
          and then
          $${bf r}_stimes{bf r}_t=(ldots,ldots,ldots) .$$
          The area is then finally given as
          $${rm area}(F)=int_0^1int_0^{2pi}bigl|{bf r}_stimes{bf r}_tbigr|>dt>ds .$$
          The resulting integral will be simpler than dreaded.






          share|cite|improve this answer














          Since in the end you have to integrate over the unit disc in the $(x,y)$-plane your source proposes to use polar coordinates instead of $x$ and $y$ as parameters. Then $x=scos t$, $y=ssin t$. In this way the idea $z=3xy$ $>(x^2+y^2leq1)$ translates into
          $${bf r}(s,t)=(scos t,ssin t,3s^2cos tsin t)qquad(0leq sleq 1, 0leq tleq2pi) .$$
          In order to find the area of this floppy disc $F$ we have to compute
          $${bf r}_s=(cos t,sin t, 6scos tsin t),quad {bf r}_t=bigl(-ssin t,scos t ,3s^2cos(2t)bigr)$$
          and then
          $${bf r}_stimes{bf r}_t=(ldots,ldots,ldots) .$$
          The area is then finally given as
          $${rm area}(F)=int_0^1int_0^{2pi}bigl|{bf r}_stimes{bf r}_tbigr|>dt>ds .$$
          The resulting integral will be simpler than dreaded.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 19 at 13:37

























          answered Nov 19 at 10:50









          Christian Blatter

          171k7111325




          171k7111325












          • Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
            – Lorenzo B.
            Nov 22 at 18:44




















          • Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
            – Lorenzo B.
            Nov 22 at 18:44


















          Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
          – Lorenzo B.
          Nov 22 at 18:44






          Sorry, it's me again. Shouldn't you integrate $|bf r_s times bf r_t|{f}(bf r)$ instead of just $|bf r_s times bf r_t|$?
          – Lorenzo B.
          Nov 22 at 18:44




















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          draft discarded




















































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