The given curve is rotated about the -axis. Find the area of the resulting surface. y=14x2−12ln⁡x,1≤x≤2

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Terry Ray · Accepted Answer

Step 1  Sy=∫ab2πx(dx)2+(dy)2  We will use the following version of the formula  Sy=∫ab2πx1+(dydx)2dx  The equation of the curve is y=14x2−12ln⁡x  Differentiate both sides with respect to x  dydx=14⋅2x2−1−12⋅1x  dydx=x2−12x  Substitute this in the formula for Sy, to get  Sy=∫122πx1+[x2−12x]2dx  Sy=∫122πx1+[x2]2+[12x]2−2[x2][12x]dx  Pay attention to the red term  Sy=∫122πx1+[x2]2+[12x]2−12dx  On simplification, the red term reduces to −12  Now, we will combine this with 1 that is already present inside the square root, so that −12 now becomes 12.  Step 2  Sy=∫122πx[x2]2+[12x]2+12dx  But, we know that 12 can be rewritten as

Corgnatiui · Accepted Answer

Step 1  y=14x2−12ln⁡x (1)  where f is positive and has a continuous derivative, we define the surface area of the surface obtained by rotating the curve y=(x),a≤x≤b, about the y-axis as  s=∫ab2πx1+(dydx)2dx  Step 2  differentiate (1) with respect to x  dydx=12x−12x  (dxdy)2=(12x−12x)2  (dxdy)2=(14x2−12+14x2)  s=∫122πx1+14x2−12+14x2dx  s=∫122πx14x2+12+14x2dx  =∫122πx(12x+12x)2dx  =∫122πx(12x+12x)dx  =π∫122(x2+1)dx  =π[13x2+x]12  =π(83+2−13−1)  The area of the surface

star233 · Accepted Answer

(y=frac{1}{2}x-frac{1}{2x}

The given curve is rotated about the -axis. Find the

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