Final answer:
To find the volume of the solid obtained by rotating the region bounded by x=6-y, y=0, and x=0 about the y-axis, set up the integral ∫ (from y=0 to y=6) 2π(6 - y)(y) dy using the cylindrical shell method, without evaluating it.
Step-by-step explanation:
To set up the integral for the volume of the solid obtained by rotating the region bounded by the curves x = 6 - y, y = 0, x = 0 about the y-axis, we will use the method of cylindrical shells. This method involves integrating the circumference of each cylindrical shell multiplied by its height and thickness.
The formula for the volume using the cylindrical shell method is:
∫ 2πrh dy
Where r is the radius from the y-axis to the shell (which is equal to x), h is the height of the shell (which depends on the curve x = 6 - y in this case), and dy is the thickness of the shell.
Using the given equations, we can express the integral as:
∫ (from y=0 to y=6) 2π(6 - y)(y) dy
This integral setup reflects the volume of the solid formed by rotating the given region about the y-axis, but it is not evaluated.