Explanation Given information: The function, y = sin 5 x Concept used: Area under the graph of y = f ( x ) from x = a to x = b is given by the definite integral, ∫ a b f ( x ) d x sin 2 x = 1 − cos 2 x Calculation: We know that the area under the graph of y = f ( x ) from x = a to x = b is given by the definite integral, ∫ a b f ( x ) d x . So using this concept the area under the graph of y = sin 5 x from x = 0 to x = π is given by the definite integral A = ∫ 0 π sin 5 x d x Now first we will compute the indefinite integral by substitution method. So we get ∫ sin 5 x d x = ∫ sin 4 x ⋅ sin x d x = ∫ ( sin 2 x ) 2 ⋅ sin x d x = ∫ ( 1 − cos 2 x ) 2 ⋅ sin x d x Substitute cos x = t ⇒ − sin x d x = d t ⇒ sin x d x = − d t So with these substitutions, above indefinite integral becomes ∫ sin 5 x d x = ∫ ( 1 − cos 2 x ) 2 ⋅ sin x d x = ∫ ( 1 − t 2 ) 2 ( − d t ) = − ∫ ( 1 − 2 t 2 + t 4 ) d t = − ( t − 2 3 t 3 + 1 5 t 5 ) = − ( cos x − 2 3 cos 3 x + 1 5 cos 5

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Question

Chapter 7.2, Problem 8E

To determine

To compute the area under the graph of y=sin5x from x=0 to x=π

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Chapter 7.2, Problem 7E

Chapter 7.2, Problem 9E

Chapter 7 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

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To compute the area under the graph of y = sin 5 x from x = 0 to x = π

Solution Summary: The author explains that the area under the graph of y=mathrmsin5x is given by the definite integral.

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To compute the area under the graph of y=sin5x from x=0 to x=π

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Chapter 7 Solutions