(a) Explanation The equation of ellipse is, x 2 a 2 + y 2 b 2 = 1 (1) The value of x and y are x = a cos t and y = b sin t . Substitute a cos t for x and b sin t for y in the equation (1). x 2 a 2 + y 2 b 2 = 1 ( a cos t ) 2 a 2 + ( b sin t ) 2 b 2 = 1 Differentiate the value of x and y with respect to t . x = a cos t d x d t = − a sin t y = b sin t d y d t = b cos t Calculate the value of ( d x d t ) 2 + ( d y d t ) 2 . ( d x d t ) 2 + ( d y d t ) 2 = ( − a sin t ) 2 + ( b cos t ) 2 = a 2 sin 2 t + b 2 cos 2 t = a 2 ( 1 − cos 2 t ) + b 2 cos 2 t = a 2 − a 2 cos 2 t + b 2 cos 2 t = a 2 + cos 2 t ( b 2 − a 2 ) Substitute − c 2 for b 2 − a 2 in the above equation. ( d x d t ) 2 + ( d y d t ) 2 = a 2 + cos 2 t ( b 2 − a 2 ) = a 2 + cos 2 t ( − c 2 ) = a 2 − c 2 cos 2 t Calculate the surface area of the ellipse about major axis. S = ∫ a b 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t Substitute b sin t for y and a 2 − c 2 cos 2 t for ( d x d t ) 2 + ( d y d t ) 2 in the above equation. S = ∫ a b 2 π y ( d x d t ) 2 + ( d y d t ) 2 d t S = ∫ − π 2 π 2 2 π ( b sin t ) a 2 − c 2 cos 2 t d t S = 2 ∫ 0 π 2 2 π ( b sin t ) a 2 − c 2 cos 2 t d t (2) Take u = c cos t . Differentiate the above equation with respect to t . u = c cos t d u = − c sin t d t d t = − d u c sin t Find the limits as below (b) To determine The surface area of the ellipse by rotating about its minor axis.

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ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.5, Problem 64E

(a)

To determine

The surface area of the ellipse by rotating about its major axis.

(b)

To determine

The surface area of the ellipse by rotating about its minor axis.

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Chapter 10.5, Problem 63E

Chapter 10.5, Problem 65E

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The surface area of the ellipse by rotating about its minor axis.

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