Explanation Given: The solid region is formed by the equation of par of the ellipse x 2 + 2 y 2 = 4 , first quadrant and x ≥ 0 , then these curve is rewrite in terms of x is y = 4 − x 2 2 and y = − 4 − x 2 2 . The region R is bounded by the above curves and first quadrant. Let f ( x ) = 4 − x 2 2 and g ( x ) = − 4 − x 2 2 . Formula used: The volume of the solid (shell method) is V = ∫ a b 2 π x ( f ( x ) − g x ) d x , where f ( x ) and g ( x ) is mentioned above. Calculation: Let l = f ( x ) − g ( x ) . Calculate the value of l with f ( x ) = 4 − x 2 2 and g ( x ) = − 4 − x 2 2 . l = ( 4 − x 2 2 ) − ( − 4 − x 2 2 ) = 2 4 − x 2 2 To find the limit of the integration, it is enough to find the intersection points of the curve f ( x ) = 4 − x 2 2 and g ( x ) = − 4 − x 2 2 as follows, 4 − x 2 2 = − 4 − x 2 2 2 4 − x 2 2 = 0 4 − x 2 2 = 0 4 − x 2 2 = 0 Further simplification of the above equation is, 4 − x 2 = 0 x 2 = 4 x = 2 Substitute the value of l = 2 4 − x 2 2 in the above mentioned formula and take x = 0 as lower limit and x = 2 as upper limit to calculate the volume of the solid

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Author: Briggs

Publisher: PEARSON

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Chapter 6.4, Problem 31E

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To find: The volume of the solid generated when R is revolved about the y axis.

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Chapter 6.4, Problem 30E

Chapter 6.4, Problem 32E

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The volume of the solid generated when R is revolved about the y axis.

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