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I have to find area S of $x^2=ay,x^2=by,x^3=cy^2,x^3=dy^2(0<a<b,0<c<d)$. It is difficult for me. Can someone help me to sovle it. Thankyou very much

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  • $\begingroup$ Can you clarify the question? It seems you want the area bounded by four curves but after graphing I think it should be you have two different areas, each bounded between two curves from $x=0$ to $x=p$ where $p$ is a positive real number. $\endgroup$
    – John Molokach
    Jun 1, 2016 at 14:54

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Interesting question! I would start by drawing a picture of the region described, and setting up the area in terms of integrals. Then, evaluate those integrals.

It seems that each of the first two curves intersect each of the last two curves in exactly one non-zero point. That gives you the four “corners” of the region described.

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  • $\begingroup$ I drew it and i can't see that it is cycle $\endgroup$
    – Tuanlee
    Jun 1, 2016 at 14:59
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    $\begingroup$ @Tuanlee: What do you mean when you say "can't see that it is cycle"? Do you mean it's not clear the curves bound a region? Maybe conditions on the constants need to be installed for that. $\endgroup$
    – Matthew Leingang
    Jun 1, 2016 at 15:03
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HINT

Substitute $\frac{x^2}{y} = u$ and $\frac{x^3}{y^2} = v$ . So $u$ goes from $a$ to $b$ and $v$ goes from $c$ to $d$. Don't forget the Jacobian. Hope this helps

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  • $\begingroup$ i will try your opinion $\endgroup$
    – Tuanlee
    Jun 1, 2016 at 15:42
  • $\begingroup$ Take a look at this (similar) question for inspiration on how to proceed with this hint: math.stackexchange.com/a/1697265/159845. $\endgroup$
    – StackTD
    Jun 1, 2016 at 15:49
  • $\begingroup$ @StackTD thankyou very much $\endgroup$
    – Tuanlee
    Jun 1, 2016 at 15:52

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