Volumes of Revolution Using Cylindrical Shells: Solving for the Volume of a Region
In this video, we explore how to find the volume of a region bounded by two functions using the method of cylindrical shells. Specifically, we revolve the region bounded by y = -x^2 + x, y = 0 about the y-axis and ALSO about the line x = – 3.
The technique of cylindrical shells is particularly useful for setting up and solving integrals in cases where the axis of rotation is parallel to the axis of the variable of integration.
What You Will Learn:
How to visualize the region bounded by two curves and its revolution around an axis.
Setting up the formula for cylindrical shells to calculate volumes.
Solving the definite integral to compute the volume of the solid of revolution.
The method of cylindrical shells simplifies the process of finding volumes of more complex shapes formed by rotating regions around axes that do not necessarily align with the standard axes. In this example, you will see step-by-step how the shell radius and shell height are used to form the integral that computes the volume.
Watch this video to learn:
Cylindrical shells method explained step by step.
Setting up the volume integral with respect to 𝑥.
Solving for the volume of a region rotated around x=−3.
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