SOLUTION: the finite region enclosed by the curve with equation y=9-x² and the x-axis is rotated through 360º about the x-axis. find, to 3 significant figures, the volume of the solid genera

Algebra ->  Volume -> SOLUTION: the finite region enclosed by the curve with equation y=9-x² and the x-axis is rotated through 360º about the x-axis. find, to 3 significant figures, the volume of the solid genera      Log On


   

Question 162738: the finite region enclosed by the curve with equation y=9-x² and the x-axis is rotated through 360º about the x-axis. find, to 3 significant figures, the volume of the solid generated.
Answer by vleith(2983) About Me  (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
graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C+9-x%5E2%29
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 9-x%5E2 from 0 to 3. So the area of each circle is pi%2Ar%5E2 = pi%2A%28%289-x%5E2%29%29%5E2
pi%2A%28%289-x%5E2%29%29%5E2
pi%2A%2881-18x%5E2+%2B+x%5E4%29
Now we need to take that Area and integrate if over from 0 to 3
int%28+A%5Bx%5D%2C+dx%2C+0%2C+3+%29

int%28+pi%2A%2881-18x%5E2+%2B+x%5E4%29%2C+dx%2C+0%2C+3+%29
pi*(81x - 6x^3 + x^5/5) from 0 to 3
pi*(81*3 - 6*3^3 + (3^5)/5) - pi*(0+0+0)
pi%2A%28243+-+162+%2B+48.6%29
pi%2A129.6
Remember to double it to pick up the volume on the left side of the y-axis
2%2Api%2A129.6
814.3
814 to three sig figs