The shell method is a mathematical technique used to find the volume of a solid of revolution, where the solid is created by rotating a 2-dimensional shape around a given axis.
The equation for finding volume using the shell method is V = 2π∫abx·f(x) dx, where a and b are the limits of integration, x is the variable of integration, and f(x) is the function representing the cross-sectional area of the solid at a given value of x.
To find the limits of integration, you must first determine the range of values for the variable of integration (in this case, x) by solving the equation x = 3y - y^2 for y. Then, you can use these values to determine the limits of integration for x.
The shell method and the disk method are both techniques used to find the volume of solids of revolution. However, the main difference is that the shell method integrates along the axis of revolution, while the disk method integrates perpendicular to the axis of revolution.
The shell method has many real-life applications, such as finding the volume of a wine barrel or a water tower, determining the amount of paint needed to coat a cylindrical object, or calculating the volume of a human organ based on medical imaging data.