Feodalherren said:Homework Statement
Find the volume of the region bounded by the curve y=x^(1/3) and y=x rotated about the line y=1.
Homework Equations
The Attempt at a Solution
My teacher's solution is 4∏/15 . I got 11∏/210.
http://imageshack.us/a/img443/426/0zhs.jpg
http://imageshack.us/a/img443/426/0zhs.jpg
Where am I going wrong?
The formula for finding the volume of a solid using shells is V = ∫ab 2πrh(x)dx, where r is the radius of the shell and h(x) is the height of the solid at a given point x.
The limits of integration for the shell method are determined by finding the points of intersection between the curves that define the solid and the axis of rotation. These points will be used as the upper and lower limits of integration.
No, the shell method can only be used to find the volume of solids that can be rotated around an axis. This method is commonly used for finding the volume of objects such as cylinders, cones, and spheres.
The disk method and the shell method are both used to find the volume of solids, but they differ in the shape of the cross-sections used for integration. The disk method uses circles as the cross-sections, while the shell method uses cylindrical shells.
One limitation of the shell method is that it can only be used for solids with a simple axis of rotation. If the axis of rotation is more complex, it may be necessary to use other methods, such as the disk method or the washer method, to find the volume of the solid.