Answer:
In order to find the surface area obtained by rotating the curve we need to find surface area for rotation along x axis then substituting values in the integral.
Given: y=3x, for x is in [1,125] about the y axis.
y=x³ ,0≤x≤2
Lets find the surface area of the rotating curve,
Finding surface area for rotation along x-axis,
s=2π∫ᵇₐ y√1+(y)² dx
y=x³
y'=3x²
By rotating the curve y=x³ about the x axis in the interval [0,2]
s=2π∫₀²(x³)√1+(3x²)² dx
Assuming u =1+9x⁴
du=36x³dx
=dx/36x³
Substituting respective values of u and du in the integral,
s= 2π∫₀²(x³)√u du/36x³
s= 2π∫₀²√u du
s= 2π/36.2/3[u.3/2]₀²
s= π/27[(145)³/²-(1)³/²]
s= π/27 [1746.03-1]
s= π/27 [1745.03]
s= 64.67π
s= 64.67(3.14)
s=203.06 square units.
hence, the surface area is 203.06 square units.
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