Final answer:
To find the volume generated by rotating the region bounded by the given curves about the y-axis, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume generated by rotating the region bounded by the curves y=9e^(-x^2), y=0, x=0, and x=1 about the y-axis, we can use the method of cylindrical shells. First, we need to determine the height of the cylindrical shells. The height can be found by subtracting the y-coordinate of the lower curve from the y-coordinate of the upper curve for each x-value. In this case, the upper curve is y=9e^(-x^2) and the lower curve is y=0.
Next, we need to determine the radius of each cylindrical shell. The radius is the distance from the y-axis to the x-value. Since we are rotating the region about the y-axis, the radius is simply the x-value.
Finally, we integrate the volume of each cylindrical shell from x=0 to x=1 using the formula V = 2πx(height)(radius). The integral can be set up as follows: V = ∫(0 to 1) 2πx(9e^(-x^2))(x) dx.