Final answer:
To calculate the volume of the solid generated by revolving the region bounded by y=9x², y=3, and x=0 in quadrant I about the y-axis, sketch the curves, find their intersection, and apply the disk method to set up and evaluate the appropriate integral.
Step-by-step explanation:
To find the volume of the solid generated by revolving the specified region about the y-axis, let's identify the steps needed:
First, sketch the curves y=9x², y=3, and x=0 to understand the region bounded by them in quadrant I of the coordinate plane. Since y=9x² represents a parabola that opens upwards and x=0 represents the y-axis, our region is the area under the parabola and the horizontal line y=3 until these curves intersect.
Next, find the points of intersection by setting 9x² equal to 3, which yields x=±√(⅓). Because we are considering quadrant I, we only take the positive square root, x=√(⅓).
To revolve the region around the y-axis, apply the disk method or the shell method. Both approaches will require setting up an integral with appropriate limits to calculate the volume.
In the case of the disk method, integrate pi*(radius)² with respect to y, from 0 to 3, where the radius is 'x'. Since y=9x², x can be expressed as √(y/9).
The integral to find the volume V using the disk method becomes V = ∫_0^3 π[√(y/9)]² dy, simplify and evaluate this integral.
This comprehensive approach helps visualize the region and mathematically determine the volume of the solid formed by rotation about the y-axis.