Volume of Revolution Disk Method
Determine the volume of the solid generated by rotating the function about the y-axis on
The disk and washer methods are powerful techniques for finding the volume of a solid created by rotating a curve around an axis. In this problem, we're revolving the function around the y-axis, which means we are interested in how the shape formed by the function creates a 3D solid as it rotates.
Conceptually, the disk method involves slicing the solid into thin, flat circular disks that are stacked on top of each other along the axis of rotation. Each disk represents a small section of the solid. The radius of each disk corresponds to the distance from the axis of rotation (in this case, the y-axis) to the function. The thickness of each disk is determined by how much the function changes over a small interval. By calculating the volume of each disk and adding them up, we get the total volume of the solid.
When we use the washer method, we account for situations where the shape being revolved has a hollow center, but in this case, because we are rotating around the y-axis and there’s no hollow section, we are essentially using the disk method.
In this specific problem, we imagine rotating the curve about the y-axis, creating disks that extend horizontally from the y-axis to the function. To calculate the total volume of the solid, we add up the volumes of these disks over the entire interval, from zero to four. This summation, handled using integration, gives us the full volume of the solid formed by the rotation.
This method is useful because it allows us to break down complex 3D shapes into simpler 2D slices, making it easier to calculate the volume through calculus.