so you draw out the region, shade it, and then imagine it as being pasted onto a glass door, where the hinges of the door is the Y axis and spin the door.
The shape ends up looking a little like the red piece on the right side.
The region between the curve (y=1/x 2) and the x-axis from x=1/2 to x=2 is revolved about the y-axis to generate a solid. Find the volume of the solid.
I know the answer is pi ln16
i tried drawing it out but i am confused witht the "revolved about the Y-AXIS" i don't see it. i see x-axis though...
No, that is the wrong integral. That would be the volume if the region were rotated around the x axis. Since you are rotating around the y-axis, a small segment above the x-axis, with height y= and thickness dx, rotated around the y-axis, will form a ring with , the circumference of the circle, height and thickness dx and so volume . The entire volume is the "sum" of those which becomes the integral
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