Final answer:
To calculate the volume of a solid formed by rotation, use calculus to integrate the area of a cross-section (often circular) along the axis of rotation. The volume can be simpler geometric shapes multiplied by their height for cylindrical shapes. More complex shapes may be approximated using standard geometric volumes and dimensions.
Step-by-step explanation:
To find the volume of a solid by rotating a region bounded by curves about a specified line, you employ methods of calculus, typically integrating a cross-sectional area along the axis of rotation. The process includes finding the limits of integration (which correspond to where the region starts and ends along the axis of rotation) and setting up the integral with the correct formula for the cross-sectional area.
For cylindrical shapes, it is sometimes easier to think of the problem in simpler geometric terms. The volume of a cylinder is the cross-sectional area times the height (V = Ah). When the cross-section is a circle, the area is πr², where r is the radius. In the case of rotation, the height in the formula corresponds to the thickness of a small slice of the solid, which is represented by dx or dy in the integral, depending on the axis of rotation.
The process is similar to estimating the volume of more complex shapes by using standard geometric shapes as models, like spheres or cubes, to get linear dimensions (radius, length, width, height) and then applying those dimensions to standard geometric formulas to calculate volume. For example, if you know the volume of a cone is (1/3)πr²h, where r is the radius of the base and h is the height, this knowledge can be used to set up an integral for a solid that resembles a cone when rotated about an axis.