Answer to Question #346587 in Calculus for Likhu

Question #346587

Find the perimeter of loop of the curve 3ay²=x²(a-x)

1
Expert's answer
2022-06-01T03:35:11-0400

3ay2=x2(ax),3ay^2=x^2(a-x),

y2=x2(ax)3a,y^2=\frac{x^2(a-x)}{3a},

y=±xax3a.y=\pm \frac{x\sqrt{a-x}}{\sqrt{3a}}.

x=0: y=0;

x=a: y=0.

The curve is symmetrical about the x-axis and meets it in points (0, 0) and (a, 0), so the loop lies between this points:



To find the perimeter of the loop, we can double its part above the x-axis.

P=20a1+(dydx)2dx.P=2 \int_0^a \sqrt{1+(\frac{dy}{dx})^2}dx.

dydx=ddx(xax3a)=13a(axx2ax)=\frac{dy}{dx}=\frac{d}{dx}(\frac{x\sqrt{a-x}}{\sqrt{3a}})=\frac{1}{\sqrt{3a}}(\sqrt{a-x}-\frac{x}{2\sqrt{a-x}})=

=13a2(ax)x2ax=2a3x23aax.=\frac{1}{\sqrt{3a}}\frac{2(a-x)-x}{2\sqrt{a-x}}=\frac{2a-3x}{2\sqrt{3a}\sqrt{a-x}}.


1+(dydx)2=1+(2a3x23aax)2=1+(\frac{dy}{dx})^2=1+(\frac{2a-3x}{2\sqrt{3a}\sqrt{a-x}})^2=

=1+(2a3x)212a(ax)=12a212ax+4a212a+9x212a(ax)==1+\frac{(2a-3x)^2}{12a(a-x)}=\frac{12a^2-12ax+4a^2-12a+9x^2}{12a(a-x)}=

=16a224ax+9x212a(ax)=(4a3x)212a(ax).=\frac{16a^2-24ax+9x^2}{12a(a-x)}=\frac{(4a-3x)^2}{12a(a-x)}.


So we have

P=20a(4a3x)212a(ax)dx=223a0a4a3xaxdx=13a0a4a3xaxdxP=2 \int_0^a \sqrt{\frac{(4a-3x)^2}{12a(a-x)}}dx=\frac{2}{2\sqrt{3a}} \int_0^a \frac{4a-3x}{\sqrt{a-x}}dx= \frac{1}{\sqrt{3a}} \int_0^a \frac{4a-3x}{\sqrt{a-x}}dx

Substitution: ax=ua-x=u , du=dx,du=-dx, x=au.x=a-u.

x=0:u=ax=a0=a;x=0: u=a-x=a-0=a;

x=a:u=ax=aa=0.x=a: u=a-x=a-a=0.


P=13a0a4a3xaxdx=13aa04a3(au)udu=P=\frac{1}{\sqrt{3a}} \int_0^a \frac{4a-3x}{\sqrt{a-x}}dx=-\frac{1}{\sqrt{3a}} \int_a^0\frac{4a-3(a-u)}{\sqrt{u}}du= s

=13aa0a+3uudu=13aa0(au+3u)du==-\frac{1}{\sqrt{3a}} \int_a^0\frac{a+3u}{\sqrt{u}}du=-\frac{1}{\sqrt{3a}} \int_a^0 (\frac{a}{\sqrt{u}}+3\sqrt{u})du=

=13a(2au+2uu)a0==-\frac{1}{\sqrt{3a}} (2a\sqrt{u}+2u\sqrt{u})|_a^0=

13a[(2a0+200)(2aa+2aa)]=-\frac{1}{\sqrt{3a}}[ (2a\sqrt{0}+2\cdot 0 \cdot \sqrt{0})- (2a\sqrt{a}+2a\sqrt{a})]=

=4aa3a=4a3.=\frac{4a\sqrt{a}}{\sqrt{3a}}=\frac{4a}{\sqrt{3}}.


Answer: 4a3.\frac{4a}{\sqrt{3}}.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment