3ay2=x2(a−x),
y2=3ax2(a−x),
y=±3axa−x.
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=2∫0a1+(dxdy)2dx.
dxdy=dxd(3axa−x)=3a1(a−x−2a−xx)=
=3a12a−x2(a−x)−x=23aa−x2a−3x.
1+(dxdy)2=1+(23aa−x2a−3x)2=
=1+12a(a−x)(2a−3x)2=12a(a−x)12a2−12ax+4a2−12a+9x2=
=12a(a−x)16a2−24ax+9x2=12a(a−x)(4a−3x)2.
So we have
P=2∫0a12a(a−x)(4a−3x)2dx=23a2∫0aa−x4a−3xdx=3a1∫0aa−x4a−3xdx
Substitution: a−x=u , du=−dx, x=a−u.
x=0:u=a−x=a−0=a;
x=a:u=a−x=a−a=0.
P=3a1∫0aa−x4a−3xdx=−3a1∫a0u4a−3(a−u)du= s
=−3a1∫a0ua+3udu=−3a1∫a0(ua+3u)du=
=−3a1(2au+2uu)∣a0=
−3a1[(2a0+2⋅0⋅0)−(2aa+2aa)]=
=3a4aa=34a.
Answer: 34a.
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