Use the method of cylindrical shells to find the volume...

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the $y$ -axis.
$$y=x^{2}, \quad 0 \leqslant x \leqslant 2, \quad y=4, \quad x=0$$

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Key Concepts

Cylindrical Shell Method

The cylindrical shell method is an integration technique used to calculate the volume of a solid of revolution by summing the volumes of infinitesimally thin cylindrical shells. Instead of slicing the solid perpendicular to the axis of rotation, this method slices parallel to the axis, which can simplify calculations when the function is easier to express in terms of the variable perpendicular to the axis of rotation.

Volume of Revolution

Volume of revolution refers to the volume generated when a two-dimensional area is rotated around an axis. This concept is fundamental in problems involving integration techniques such as the disk, washer, or cylindrical shell methods, each providing a systematic approach to summing the infinitesimal volume elements that make up the solid.

Setting Up the Integral

Setting up the integral for the cylindrical shell method involves identifying the correct variable of integration, determining the limits of integration based on the region's boundaries, and expressing the dimensions of the shells. Specifically, the radius of each shell is the distance to the axis of rotation, and the height is typically defined by the difference between the given functions or boundaries.

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