Find the volume of the solid obtained by rotating the region under the graph of

Find the volume of the solid obtained by rotating the

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9.(1 point)Find the volume of the solid obtained by rotating the regionunder the graph of the functionf(x) =4x-x2about thex-axisover the interval[0,4].V= 2
Volume= Solution: SOLUTION For f ( x ) = cos ( 4 x 2 ) Volume = 2 π Z π 4 0 x · f ( x ) dx = π Z π 4 0 2 x cos ( 4 x 2 ) dx = π " sin ( 4 x 2 ) 4 # π 4 0 = π 4 2 The volume of the solid revolved around the y -axis is π 4 2 . Solution: (click on image to enlarge) Which of the following integrals represents the volume ofthe solid obtained by rotating the region bounded by the curvesx2-y2=7 andx=4 about the liney=4?Z472π(y-4)(4-py 12.(1 point) Match the following integrals with the solidwhose volume it represents. A. 2 π Z 3 0 x 5 dx B. 2 π Z 2 0 y 1 + y 2 dy C. 2 π Z 1 0 ( 3 - y )( 1 - y 2 ) dy D. 2 π Z π / 4 0 ( π - x )( cos ( x ) - sin ( x )) dx 1. The volume of the solid obtained by rotating the region bounded by x = y 2 , x = 1 , and y = 0 about the line y = 3, using cylindrical shells. 2. The volume of the solid obtained by rotating the region described by 0 y 2 , 0 x 1 1 + y 2 , about the x -axis, using cylindrical shells. 3. The volume of the solid obtained by rotating the region described by 0 x 3 , 0 y x 4 , about the y -axis, using cylindrical shells. 4. The volume of the solid obtained by rotating the region bounded by 0 x π / 4 and sin ( x ) y cos ( x ) about the line x = π , using cylindrical shells.

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