Explanation Given: Consider a function, y = h ( x ) which is continuous on [ α , β ] . The area bounded by the graph of h and the x -axis is, ∫ a b h ( x ) d x = ∫ α β y d x = ∫ a b g ( t ) f ′ ( t ) d t Here, x = f ( t ) , y = g ( t ) , for α ≤ t ≤ b is the parametric equations. Calculation: The given parametric equation is x = cos 3 t , y = sin 3 t for 0 ≤ t ≤ 2 π . Here, x = f ( t ) = cos 3 t and y = g ( t ) = sin 3 t . Differentiate f ( t ) with respect to t as follows. f ′ ( t ) = d d t ( cos 3 t ) = 3 cos 2 t d d t ( cos t ) = − 3 cos 2 t sin t Substitute 0 for t in x = cos 3 t . x = cos 3 t = cos 3 ( 0 ) = 1 Substitute π 2 for t in x = cos 3 t . x = cos 3 t = cos 3 ( π 2 ) = 0 Thus, the interval corresponding to the first quadrant is 0 ≤ x ≤ 1 . Initially, obtain the area bounded by the astroid in the first quadrant and then multiply it with 4 to obtain the complete area of the region bounded by the astroid as follows. Notice that, for first quadrant 0 ≤ t ≤ π 2 . Substitute f ′ ( t ) = − 3 cos 2 t sin t and g ( t ) = sin 3 t in ∫ α β y d x = 4 ∫ a b g ( t ) f ′ ( t ) d t . ∫ α β y d x = 4 ∫ a b g ( t ) f ′ ( t ) d t = − 4 ∫ π 2 0 sin 3 t ⋅ 3 cos 2 t sin t d t = 12 ∫ 0 π 2 sin 4 t cos 2 t d t = 12 8 ∫ 0 π 2 ( 1 − cos 2 t ) 2 ( 1 − cos 2 t ) d t On further simplifications, ∫ α β y d x = 3 2 ∫ 0 π 2 ( 1 − cos 2 t − cos 2 2 t + cos 3 2 t ) d t = 3 2 ∫ 0 π 2 ( 1 − cos 2 t − 1 + cos 4 t 2 + ( 1 − sin 2 2 t ) ( cos 2 t ) ) d t = 3 2 ∫ 0 π 2 ( 1 − cos 2 t − 1 2 − cos 4 t 2 + cos 2 t − sin 2 2 t cos 2 t ) d t = 3 2 ∫ 0 π 2 ( 1 2 − cos 4 t 2 − sin 2 2 t cos 2 t ) d t Further simplified as follows. ∫ α β y d x = 3 2 ∫ 0 π 2 ( 1 2 − cos 4 t 2 − sin 2 2 t cos 2 t ) d t = 3 4 ∫ 0 π 2 d t − 3 4 ∫ 0 π 2 cos 4 t d t − 3 2 ∫ 0 π 2 sin 2 2 t cos 2 t d t = I 1 − I 2 − I 3 Obtain the value of I 1 as follows. I 1 = 3 4 ∫ 0 π 2 d t = 3 4 [ t ] 0 π 2 = 3 4 [ π 2 − 0 ] = 3 π 8 Obtain the value of I 2 as follows

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.1, Problem 105E

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The area of the region bounded by the astroid x=cos3t,y=sin3t for 0≤t≤2π.

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Chapter 12.1, Problem 104E

Chapter 12.1, Problem 106E

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Calculus, Single Variable: Early Transcendentals (3rd Edition)

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The area of the region bounded by the astroid x = cos 3 t , y = sin 3 t for 0 ≤ t ≤ 2 π .

Solution Summary: The author calculates the parametric equations for the region bounded by the astroid x=mathrmcos3t,y=

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The area of the region bounded by the astroid x=cos3t,y=sin3t for 0≤t≤2π.

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