Khan Academy on a Stick : Solid of revolution

Khan Academy on a Stick

Calculus

Disk method around x-axis
Finding the solid of revolution (constructed by revolving around the x-axis) using the disc method.
Generalizing disc method around x-axis
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Generalizing what we did in the last video for f(x) to get the "formula" for using the disc method around the x-axis
Disc method around y-axis
Finding the volume of a figure that is rotated around the y-axis using the disc method
Disc method (washer method) for rotation around x-axis
Finding the volume of a solid of revolution that is defined between two functions
Generalizing the washer method
Looking at the example from the last video in a more generalized way
Disc method rotation around horizontal line
Solid of revolution constructing by rotating around line that is not an axis
Washer method rotating around non-axis
Washer method when rotating around a horizontal line that is not the x-axis
Part 2 of washer for non axis rotation
Doing some hairy algebra and arithmetic to evaluate the definite integral from the last video
Disc method rotating around vertical line
Volume of solid created by rotating around vertical line that is not the y-axis using the disc method.
Calculating integral disc method around vertical line
Let's calculate the integral from the last video.
Washer or ring method for vertical line rotation
Setting up the definite integral for the volume of a solid of revolution around a vertical line using the "washer" or "ring" method.
Evaluating integral for washer method around vertical line
Evaluating integral set up in the last video using washer method.

Disc method

You know how to use definite integrals to find areas under curves. We now take that idea for "spin" by thinking about the volumes of things created when you rotate functions around various lines. This tutorial focuses on the "disc method" and the "washer method" for these types of problems.

Shell method for rotating around vertical line
Introducing the shell method for rotation around a vertical line.
Evaluating integral for shell method example
Evaluating the definite integral set up using the shell method
Shell method for rotating around horizontal line
Example showing how to find the volume of a solid of revolution (constructed by rotating around the x-axis) using the shell method (this could have been done with the disk method as well).
Shell method with two functions of x
Using the shell method to rotate around a vertical line.
Calculating integral with shell method
Evaluating integral set up with shell method for two functions.
Shell method with two functions of y
Part 2 of shell method with 2 functions of y

Shell method

You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.

Disc method: function rotated about x-axis
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Figuring out the volume of a function rotated about the x-axis.
Disc method (rotating f(x) about x axis)
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The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis
Volume of a sphere
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Figuring out the equation for the volume of a sphere.
Disc method with outer and inner function boundaries
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More volumes around the x-axis.
Shell method to rotate around y-axis
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Use the "shell method" to rotate about the y-axis
Disk method: rotating x=f(y) around the y-axis
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Using the disk method around the y-axis.
Shell method around a non-axis line
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Taking the revolution around something other than one of the axes.
Shell method around a non-axis line 2
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The last part of the problem in part 7

Solid of revolution volume

Using definite integration, we know how to find the area under a curve. But what about the volume of the 3-D shape generated by rotating a section of the curve about one of the axes (or any horizontal or vertical line for that matter). This in an older tutorial that is now covered in other tutorials. This tutorial will give you a powerful tool and stretch your powers of 3-D visualization!