fa07hw7 - 5 If x = 3 sin 2 t y = 3 cos 2 t ≤ t ≤ π 3...

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Unformatted text preview: 5. If x = 3 sin 2 t , y = 3 cos 2 t , ( ≤ t ≤ π/ 3 ) , then x 2 + y 2 = 9. This is part of a circle. y x t = π 3 t = x 2 + y 2 = 9 Fig. 2-5 7. If x = 3 sin π t , y = 4 cos π t , (- 1 ≤ t ≤ 1 ) , then x 2 9 + y 2 16 = 1. This is an ellipse. y x t = t = 1 t =- 1 x 2 9 + y 2 16 = 1 Fig. 2-7 9. If x = cos 3 t , y = sin 3 t , ( ≤ t ≤ 2 π) , then x 2 / 3 + y 2 / 3 = 1. This is an astroid. y t = 3 π/ 2 t = 2 π t = t = π/ 2 x 2 / 3 + y 2 / 3 = 1 t = π Fig. 2-9 21. y = x 3 , 0 ≤ x ≤ 1. ds = √ 1 + 9 x 4 dx . The area of the surface of rotation about the x-axis is S = 2 π integraldisplay 1 x 3 radicalbig 1 + 9 x 4 dx Let u = 1 + 9 x 4 du = 36 x 3 dx = π 18 integraldisplay 10 1 √ u du = π 27 ( 10 3 / 2- 1 ) sq. units. 23. If y = x 3 / 2 , 0 ≤ x ≤ 1, is rotated about the y-axis, the surface area generated is S = 2 π integraldisplay 1 x radicalbigg 1 + 9 x 4 dx Let u = 1 + 9 x 4 du = 9 4 dx = 32 π 81 integraldisplay 13 / 4 1 ( u- 1 ) √ u du = 32 π 81 parenleftbigg...
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This homework help was uploaded on 02/07/2008 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell.

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fa07hw7 - 5 If x = 3 sin 2 t y = 3 cos 2 t ≤ t ≤ π 3...

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