Use the method of cylindrical shells to find the volume generated by rotating th

Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = 4x^2, y = 24x ? 8x^2

Explanation / Answer

using the Shell Method: about y-axis height ===> 24x - 8x^2 - 4x^2 radius ====> x limits: 24x - 8x^2 = 4x^2 0 = 4x^2 - 24x + 8x^2 0 = 12x^2 - 24x 0 = x^2 - 2x 0 = x * (x - 2) ----> x = 0 & 2 2 ? 2p * x * ( 24x - 12x^2 ) dx 0 2 ? 2p * ( 24x^2 - 12x^3 ) dx 0 . . . . . . . .. . . . .. . .2 2p * ( 8x^3 - 3x^4 ) ] . . . . . . . .. . . . .. . .0 2p * ( 8 * ( 2^3 - 0^3 ) - 3 * ( 2^4 - 0^4 ) ) ] 2p * ( 8 * ( 8 - 0 ) - 3 * ( 16 - 0 ) ) ] 2p * ( 64 - 48 ) 2p * 16 32p

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