Question

Compute the surface area obtained by rotating the curve y = 3 x , for x is in [1, 125] about the y-axis

Answer 1

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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Answer 2

Answer:

Given,

y=x³ ,0≤x≤2

We have to find the surface area of the surface by rotating the curve about the x axis.

For rotation about the x axis, the surface area formula is given by-

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

let u=1+9x⁴

du=36x³dx

dx/36x³

Substituting 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.

therefore, the exact surface area is 203.06 square units.

Learn more about surface area here:

brainly.com/question/4678941

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