Explanation Given information: The volume of the solid is bounded by the plane z + y = 3 and the cylinder x 2 + y 2 = 4 . The Equation of the plane as follows: z + y = 3 (1) Show the Equation of the cylinder as follows: x 2 + y 2 = 4 (2) Calculation: Consider the region bounded by the coordinate planes, the plane z + y = 3 and the cylinder x 2 + y 2 = 4 is R . Refer Equation (1). z + y = 3 z = 3 − y f ( x , y ) = 3 − y Find the volume of the region R using the relation. V = ∬ R f ( x , y ) d A = ∬ R ( 3 − y ) d y d x (3) Refer Equation (2). x 2 + y 2 = 2 2 The above Equation corresponds to a circle with radius of 2 centred at the origin. Consider the first octant of the above circle is the region R . The domain of the circle in the first quadrant lies in the range 0 to 2. Then, The lower and upper bound limit of integration over the region R in the x direction is 0 and 2. The range of the circle in the first quadrant lies in the range 0 to 4 − x 2 . The lower and upper bound limit of integration over the region R in the y direction is 0 and 4 − x 2 . Apply the limit of integration over the region R in the Equation (3). V = ∬ R f ( x , y ) d A = ∫ 0 2 ∫ 0 4 − x 2 ( 3 − y ) d y d x = ∫ 0 2 ( 3 y − y 2 2

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.2, Problem 60E

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Determine the volume of the solid in the first octant which is bounded by the coordinate planes.

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Chapter 14.2, Problem 59E

Chapter 14.2, Problem 61E

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Chapter 14 Solutions

University Calculus: Early Transcendentals (3rd Edition)

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