For what value of “c” volume of ellipsoid equal to $8pi$?












0












$begingroup$


The equation of ellipsoid is
$$x^2+bigg(frac{y}{2}bigg)^2+bigg(frac{z}{c}bigg)^2=1$$
I have taking the limits of integration
$$int_{0}^{1}int_{-2}^{2}int_{0}^{csqrt{1-x^2-frac{y^2}{4}}}dzdydx$$
Am I right just confused little bit. That's why I need confirmation.
The second question is how can I change the order of integration to evaluate the following




  1. $int_{0}^{4}int_{0}^{1}int_{2y}^{2} frac{4cos(x^2)}{2sqrt{z}}dxdydz$

  2. $int_{0}^{2}int_{0}^{4-x^2}int_{0}^{x} frac{sin2z}{4-z}dydzdx$


  3. $$int_{0}^{1}int_{3sqrt{z}}^{1}int_{0}^{ln3} frac{pi e^{2x} sin(pi y^2)}{y^2}dxdydz$$
    For (1)
    With y-innermost:
    Since $0leq yleq 1,$ $2yleq xleq 2$ and $0leq zleq 4$
    The bounds of $y$-are $0leq yleq frac{x}{2}$, the z limits are unaffected by $y$ and $x$,
    enter image description here
    hence
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dy dx dz.$$
    Solving this I get
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dydxdz=frac{8sin4}{3}$$
    Am I right changing in order of integration?


For (2) With y-innermost: since the limits of integration are $0leq xleq2,$ $0leq zleq 4-x^2$ and $0leq yleq x$, then I get the
$$int_{0}^{2}int_{0}^{2}int_{0}^{sqrt{4-z}}frac{sin2z}{4-z}dy dx dz$$
When I try to solve this (2) after changing the order of integration I am not still getting the integral in form of elementary functions, Can anyone help!
enter image description here
For (3) With z-innermost: I get
enter image description here$$int_{0}^{ln3}int_{0}^{1}int_{0}^{frac{y^2}{9}}frac{pi e^{2x} sin(pi y^2)}{y^2}dzdydx$$but still after changing the order of integration in (3) I can't simplified this.
Please help in solving all these four!
Thanks in advance!






calculus multivariable-calculus lagrange-multiplier multiple-integral







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Please avoid asking multiple questions in a single post.
    $endgroup$
    – Saad
    Jan 6 at 11:14










  • $begingroup$
    Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
    $endgroup$
    – Hagen von Eitzen
    Jan 6 at 11:22
















0












$begingroup$


The equation of ellipsoid is
$$x^2+bigg(frac{y}{2}bigg)^2+bigg(frac{z}{c}bigg)^2=1$$
I have taking the limits of integration
$$int_{0}^{1}int_{-2}^{2}int_{0}^{csqrt{1-x^2-frac{y^2}{4}}}dzdydx$$
Am I right just confused little bit. That's why I need confirmation.
The second question is how can I change the order of integration to evaluate the following




  1. $int_{0}^{4}int_{0}^{1}int_{2y}^{2} frac{4cos(x^2)}{2sqrt{z}}dxdydz$

  2. $int_{0}^{2}int_{0}^{4-x^2}int_{0}^{x} frac{sin2z}{4-z}dydzdx$


  3. $$int_{0}^{1}int_{3sqrt{z}}^{1}int_{0}^{ln3} frac{pi e^{2x} sin(pi y^2)}{y^2}dxdydz$$
    For (1)
    With y-innermost:
    Since $0leq yleq 1,$ $2yleq xleq 2$ and $0leq zleq 4$
    The bounds of $y$-are $0leq yleq frac{x}{2}$, the z limits are unaffected by $y$ and $x$,
    enter image description here
    hence
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dy dx dz.$$
    Solving this I get
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dydxdz=frac{8sin4}{3}$$
    Am I right changing in order of integration?


For (2) With y-innermost: since the limits of integration are $0leq xleq2,$ $0leq zleq 4-x^2$ and $0leq yleq x$, then I get the
$$int_{0}^{2}int_{0}^{2}int_{0}^{sqrt{4-z}}frac{sin2z}{4-z}dy dx dz$$
When I try to solve this (2) after changing the order of integration I am not still getting the integral in form of elementary functions, Can anyone help!
enter image description here
For (3) With z-innermost: I get
enter image description here$$int_{0}^{ln3}int_{0}^{1}int_{0}^{frac{y^2}{9}}frac{pi e^{2x} sin(pi y^2)}{y^2}dzdydx$$but still after changing the order of integration in (3) I can't simplified this.
Please help in solving all these four!
Thanks in advance!






calculus multivariable-calculus lagrange-multiplier multiple-integral







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Please avoid asking multiple questions in a single post.
    $endgroup$
    – Saad
    Jan 6 at 11:14










  • $begingroup$
    Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
    $endgroup$
    – Hagen von Eitzen
    Jan 6 at 11:22














0












0








0


1



$begingroup$


The equation of ellipsoid is
$$x^2+bigg(frac{y}{2}bigg)^2+bigg(frac{z}{c}bigg)^2=1$$
I have taking the limits of integration
$$int_{0}^{1}int_{-2}^{2}int_{0}^{csqrt{1-x^2-frac{y^2}{4}}}dzdydx$$
Am I right just confused little bit. That's why I need confirmation.
The second question is how can I change the order of integration to evaluate the following




  1. $int_{0}^{4}int_{0}^{1}int_{2y}^{2} frac{4cos(x^2)}{2sqrt{z}}dxdydz$

  2. $int_{0}^{2}int_{0}^{4-x^2}int_{0}^{x} frac{sin2z}{4-z}dydzdx$


  3. $$int_{0}^{1}int_{3sqrt{z}}^{1}int_{0}^{ln3} frac{pi e^{2x} sin(pi y^2)}{y^2}dxdydz$$
    For (1)
    With y-innermost:
    Since $0leq yleq 1,$ $2yleq xleq 2$ and $0leq zleq 4$
    The bounds of $y$-are $0leq yleq frac{x}{2}$, the z limits are unaffected by $y$ and $x$,
    enter image description here
    hence
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dy dx dz.$$
    Solving this I get
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dydxdz=frac{8sin4}{3}$$
    Am I right changing in order of integration?


For (2) With y-innermost: since the limits of integration are $0leq xleq2,$ $0leq zleq 4-x^2$ and $0leq yleq x$, then I get the
$$int_{0}^{2}int_{0}^{2}int_{0}^{sqrt{4-z}}frac{sin2z}{4-z}dy dx dz$$
When I try to solve this (2) after changing the order of integration I am not still getting the integral in form of elementary functions, Can anyone help!
enter image description here
For (3) With z-innermost: I get
enter image description here$$int_{0}^{ln3}int_{0}^{1}int_{0}^{frac{y^2}{9}}frac{pi e^{2x} sin(pi y^2)}{y^2}dzdydx$$but still after changing the order of integration in (3) I can't simplified this.
Please help in solving all these four!
Thanks in advance!






calculus multivariable-calculus lagrange-multiplier multiple-integral







share|cite|improve this question











$endgroup$




The equation of ellipsoid is
$$x^2+bigg(frac{y}{2}bigg)^2+bigg(frac{z}{c}bigg)^2=1$$
I have taking the limits of integration
$$int_{0}^{1}int_{-2}^{2}int_{0}^{csqrt{1-x^2-frac{y^2}{4}}}dzdydx$$
Am I right just confused little bit. That's why I need confirmation.
The second question is how can I change the order of integration to evaluate the following




  1. $int_{0}^{4}int_{0}^{1}int_{2y}^{2} frac{4cos(x^2)}{2sqrt{z}}dxdydz$

  2. $int_{0}^{2}int_{0}^{4-x^2}int_{0}^{x} frac{sin2z}{4-z}dydzdx$


  3. $$int_{0}^{1}int_{3sqrt{z}}^{1}int_{0}^{ln3} frac{pi e^{2x} sin(pi y^2)}{y^2}dxdydz$$
    For (1)
    With y-innermost:
    Since $0leq yleq 1,$ $2yleq xleq 2$ and $0leq zleq 4$
    The bounds of $y$-are $0leq yleq frac{x}{2}$, the z limits are unaffected by $y$ and $x$,
    enter image description here
    hence
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dy dx dz.$$
    Solving this I get
    $$int_{0}^{4}int_{0}^{2}int_{0}^{frac{x}{2}}dydxdz=frac{8sin4}{3}$$
    Am I right changing in order of integration?


For (2) With y-innermost: since the limits of integration are $0leq xleq2,$ $0leq zleq 4-x^2$ and $0leq yleq x$, then I get the
$$int_{0}^{2}int_{0}^{2}int_{0}^{sqrt{4-z}}frac{sin2z}{4-z}dy dx dz$$
When I try to solve this (2) after changing the order of integration I am not still getting the integral in form of elementary functions, Can anyone help!
enter image description here
For (3) With z-innermost: I get
enter image description here$$int_{0}^{ln3}int_{0}^{1}int_{0}^{frac{y^2}{9}}frac{pi e^{2x} sin(pi y^2)}{y^2}dzdydx$$but still after changing the order of integration in (3) I can't simplified this.
Please help in solving all these four!
Thanks in advance!






calculus multivariable-calculus lagrange-multiplier multiple-integral




calculus multivariable-calculus lagrange-multiplier multiple-integral






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 6 at 17:33







Noor Aslam

















asked Jan 6 at 11:08









Noor AslamNoor Aslam

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16012








  • 2




    $begingroup$
    Please avoid asking multiple questions in a single post.
    $endgroup$
    – Saad
    Jan 6 at 11:14










  • $begingroup$
    Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
    $endgroup$
    – Hagen von Eitzen
    Jan 6 at 11:22














  • 2




    $begingroup$
    Please avoid asking multiple questions in a single post.
    $endgroup$
    – Saad
    Jan 6 at 11:14










  • $begingroup$
    Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
    $endgroup$
    – Hagen von Eitzen
    Jan 6 at 11:22








2




2




$begingroup$
Please avoid asking multiple questions in a single post.
$endgroup$
– Saad
Jan 6 at 11:14




$begingroup$
Please avoid asking multiple questions in a single post.
$endgroup$
– Saad
Jan 6 at 11:14












$begingroup$
Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
$endgroup$
– Hagen von Eitzen
Jan 6 at 11:22




$begingroup$
Your ellipsoid volume is $2^2c^2$ times that of the unit ball ...
$endgroup$
– Hagen von Eitzen
Jan 6 at 11:22










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