The integral itself looks right, although the graph of that function is zero at ##\pi/4##.Michael_0039 said:
Oops my mistake, it is: 0 ≤ x ≤ πPeroK said:
The formula for finding the volume of a solid formed by rotation around the y=0 axis is V = π∫ab(f(x))2dx, where a and b represent the boundaries of the solid, and f(x) is the function that forms the boundary of the solid when rotated around the y=0 axis.
The main difference is that when rotating around the y=0 axis, the cross-sectional area is perpendicular to the y-axis, whereas when rotating around the x=0 axis, the cross-sectional area is perpendicular to the x-axis. This changes the limits of integration and the function used in the formula.
The most common shapes that are formed when rotating around the y=0 axis are cylinders, cones, and disks. Other shapes such as spheres, toruses, and paraboloids can also be formed depending on the function used.
The volume of a solid formed by rotation around the y=0 axis is calculated using a different formula and limits of integration compared to the volume of a solid formed by revolution around a different axis. The shape and orientation of the cross-sectional area also play a role in determining the formula and limits of integration.
Finding the volume of a solid formed by rotation around the y=0 axis can be useful in various situations, such as calculating the volume of a cylindrical tank, determining the volume of a cone-shaped object, or finding the volume of a pool with a curved bottom. It can also be used in engineering and design to calculate the volume of rotating components such as gears or turbines.