Once you know and understand the disk method, another good application of integrals to check out would be the washer method. Disk 2 Section 6: Multivariable Functions And Partial Derivatives Section 7: The Chain Rule For Partial Derivatives Section 8: The Directional Derivative. The radius can be any positive number when R is infinity, the region is usually called a punctured plane (Sarason, 2007). We know that the center of our disk will always have a y-coordinate of 0 because we rotated our function around the line \(\mathbf\)! Hopefully this has helped you with the disk method, but if there’s still a topic you’d like to learn about take a look at some of my other lessons and problem solutions about integrals. The center point of our disk is labeled (x, 0). Imagine we are trying to find the distance between the two points we labeled. You can see in the drawing above that I drew a copy of the disk we are considering down below the function. In fact, if you look at our drawing, you can see that the radius of each cylinder will simply depend on the x value where it’s sitting. The disk method is used to calculate the volume of a solid of revolution when integrating along an axis parallel to the axis of revolution.
How do we find the radius?Ĭonsidering that each cylinder will be a different size, it seems clear that the radius of each cylinder will depend on which cylinder we’re considering. Thinking back to our example of rotating \(y=x^2\) around the x-axis, let’s determine the radius and height of our cylinders. So in order to find their volumes, we should start with the volume of a cylinder, which is Clearly each disk is a very thin cylinder. Normally, this shift involves taking a sum or difference of the function along with the constant connected to the equation for the horizontal or vertical line.
Since we are trying to make this integral represent the sum of all of these disks, we need to think about the volume of each disk in particular. The green cylinder in the figure represents one of the infinitely thin disks that we are slicing the figure into. You can see in this drawing our function has been rotated around the x-axis to create a round cone-like 3-D figure.
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