Find the exact area of the surface obtained by rotating the curve about the x-ax

Question

Find the exact area of the surface obtained by rotating the curve about the x-axis. y = x^3, 0 lessthanorequalto x lessthanorequalto 2 9x = y^2 + 18, 2 lessthanorequalto x lessthanorequalto 6 y = Squareroot 1 + 4x, 1 lessthanorequalto x lessthanorequalto 5 y = Squareroot 1 + e^x, O lessthanorequalto x lessthanorequalto 1 y = sin pix, 0 lessthanorequalto x lessthanorequalto 1 y = x^3/6 + 1/2x. 1/2 lessthanorequalto y lessthanorequalto 1 x = 1/3(y^2 + 2)^3/2, 1 lessthanorequalto y lessthanorequalto 2 x = 1 + 2y^2, 1 lessthanorequalto y lessthanorequalto 2

Explanation / Answer

5)

we know So dV = y^2 dx = (x^3)^2 dx = x^6 dx
V = x^6 dx = x^7/7
Evaluating the integra l: /7 (2^7 - 0^7) = 128/7 = 128/7

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