Question

Sketch the region bounded by the curves y=6x² , y=6 and x=0 then find the volume of the solid generated by revolving this region about the x-axis.

Answer 1

To find the volume, sketch the region, determine the shape and height of the solid, and integrate using the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region about the x-axis, we need to use the method of cylindrical shells.

1. First, sketch the region bounded by the given curves. The region is a rectangle with a triangle on top.
2. Next, find the height of the rectangle by subtracting the y-values of the curves y=6 and y=6x². The height is 6 - 6x².
3. Then, find the length of the rectangle by determining the x-coordinate where the curves y=6 and y=6x² intersect. The intersection point is (1, 6).
4. Finally, integrate the expression 2πrh, where r is the x-coordinate and h is the height, over the interval [0, 1].

By following these steps, you can calculate the volume of the solid generated by revolving this region about the x-axis. The answer will be in cubic units.