EXAMPLE The arc of the parabola y = 2x2 from (2, 8) to (4, 32) is rotated about the y-axis. Find the area of the resulti

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Example The Arc Of The Parabola Y 2x2 From 2 8 To 4 32 Is Rotated About The Y Axis Find The Area Of The Resulti 1 (30.56 KiB) Viewed 18 times

EXAMPLE The arc of the parabola y = 2x2 from (2, 8) to (4, 32) is rotated about the y-axis. Find the area of the resulting surface. SOLUTION 1 Using y = 2x² and dy dx we have, from this formula, s = √2xx ds S= - IT Substituting u 1+ 16x, we have du = 16/65 31 SOLUTION 2 257 Using x= we have - 2x√/1 - 2x * x√ 1 + 16x² dx. √ and = = 2n = W 16 14 P S = [ 2mx ds = [ 2xx√/1 + (dx) dy dx 257 dx 1+ Vudu (where u = 1 + 8y) (as in Solution 1) dx. Remembering to change the limits of integration, we have