Example One
examples and exercises involving rotational solids created by rotating regions around axes. Calculate volumes and areas of shapes defined by curves, lines, and axes in calculus.
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Presentation Transcript
Example One The region between the curve ? = ?2, the x axis and the lines ? = 1,??? ? = 3 is rotated by 360 around the x axis. What is the volume of the resulting shape? Graph
Example Two: The region R is bound by the curve with equation ? = ?2 1, the x axis and the lines ? = 0 and ? = 2. a) Calculate the area of the region R. The region R is rotated 360 about the x axis. b) Find the exact volume of the solid formed.
Revolving around the ?-axis ? To revolve instead around the ?-axis, we simply swap the roles of the ? and ? axes! If the curve is revolved around ?-axis: ? = ? ?2 ?? ? ? ?? ? ? The diagram shows the curve with equation ? = ? 1. The region ? is bounded by the curve, the ?-axis and the lines ? = 1 and ? = 3. The region is rotated through 360 about the ?-axis. Find the volume of the solid generated. ? = ?2+ 1 3 ?2+ 12 ?? ? ? = ? ? 1 = =1016? 15 ? = ? 1 3 ? 1 ?
Test Your Understanding 32? + 1. The region ? is bounded by the curve, the A curve has equation ? = ?-axis and the lines ? = 2 and ? = 4. The region is rotated through 360 about the ?-axis. Find the volume of the solid generated. ?3= 2? + 1 ? =1 2?3 1 41 2 2?3 1 ? 2 ? = ? ?? 2 2 = =7715? 14
The region R is formed by the y axis, the lines y = 0 and y = 9 and the curve with equation ? = ? + 22 a) Find the area of the region R. The region R is rotated 360 about the y axis. b) Find the volume of the solid formed.
Example ? The region ? is bounded by the curve with equation ? = ?3+ 2, the line ? = 5 2? and ? and ?-axes. (a) Verify that the coordinates of ? are 1,3 . A solid is created by rotating the region 360 about the ?-axis. (b) Find the volume of this solid. ? = ?3+ 2 ? ? = 5 2? ? ? ? ? a 13+ 2 = 3, 5 2 1 = 3 Find the two volumes separately: ? ? = ?3+ 2 1 ?3+ 22 ?? = =36? ?1= ? ? 7 0 5 2? intersects the ?-axis at 2.5 ?2=1 ? b ? = 5 2? ?1 3? 32 1.5 =9? ?2 It s a cone! ? 2 ? 2.5 1 ????????=36? +9? 2=135? 7 14
Volumes by Subtraction ? The diagram shows the region ? bounded by the curves with equations ? = ? and ? = 8? and the line ? = 1. The region is rotated through 360 about the ?-axis. Find the exact volume of the solid generated. 1 1 ? = ? = ? 8? ? Do volume under top curve and subtract volume under bottom curve. ? 1 Point of intersection: 1 8?= ? =1 3 2= 1 ? 8? 4 ? 1 ?2 ?? = =15? ?1= ? 1 32 4 2 1 1 ?? = =3? ?2= ? 1 8? 64=27? 64 4 ??=15? 32 3? 64
Test Your Understanding ? 3?, The area between the lines with equations ? = ? and ? = where ? 0 is rotated 360 about the ?-axis. Determine the volume of the solid generated. 3? ? = ? = ? ? 1 3 ?3= ? ? = ? ?3 ? = 0 ? ? 1 ? + 1 = 0 Intersect at ? = 1,0,1 ? 2 ? 1 ?? = =3? 1 3 ?1= ? ? 5 0 ?2=1 ??=3? 1 =? 3=4? 15 3? 12 5 ? 3
