SOLUTION: the finite region enclosed by the curve with equation y=9-x� and the x-axis is rotated through 360� about the x-axis. find, to 3 significant figures, the volume of the solid genera

Question 162738: the finite region enclosed by the curve with equation y=9-x� and the x-axis is rotated through 360� about the x-axis. find, to 3 significant figures, the volume of the solid generated.
Answer by vleith(2983) (Show Source): You can put this solution on YOUR website!
You'll need calculus for this one. If you plot the given function you see that it is a parabola with vertex at (0,9) and x intercepts at (-3,0) and (3,0). So let's find the volume in the solid between x=0 and x=3. Then we just need to double that to get the entire volume

When you rotate that curve around the x axis, you'll get a solid that you can slice into 'disks'. Imagine the disks are standing on end. As you make the disk thinner and thinner, it approaches becoming a whole bunch of 'circles' with a width of deltax. The radius of each thin circle is given by from 0 to 3. So the area of each circle is =

Now we need to take that Area and integrate if over from 0 to 3

pi*(81x - 6x^3 + x^5/5) from 0 to 3
pi*(81*3 - 6*3^3 + (3^5)/5) - pi*(0+0+0)

Remember to double it to pick up the volume on the left side of the y-axis

to three sig figs
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