E.+Duncan

=Finding the Volume By Integration =

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In this problem, you find the area of the shaded region wrapped around the x-axis. This is a video created for the purpose of learning/teaching calculus with the help of the Beautiful Brittney and the Magnificent Mahala! This video should explain how to find a volume by integration. Enjoy!

**~ I <3 Calculus! ~ **

State the different examples that you could integrate and get the same volume.
y = -(9-x^2)^(1/2) rotated around the x-axis would give the same area, as would x = y^2+9 or x = -(y^2+9) rotated around the y-axis. -lovebk.

What happens is this curve is shifted up 4 units? What line would it have to be rotated around to obtain the same volume?
if the curve was shifted up 4 units, the area of the 2-D figure would be the same, yet if you were to rotate it around the x-axis, the volume would be much greater. To obtain the same volume, the figure would have to be rotated around y = 4. The new coordinates of the bottom of the figure would be (-3, 4) and (3,4).The range would remain the same. -lovebk.

What line or lines would the equation 9=(x-2)^2+(y+10)^2 have to be rotated around to obtain the same volume. Explain your reasoning.
it would have to be rotated around the line y= -10 because the graph of the equation shifted down 10 units. ~Eve