Volume through the disc method

In summary, the conversation discussed the disc method in Calculus 2 and its application in rotating an area around an axis to obtain a volume. It was mentioned that the method is typically used for functions rotated around the x-axis or y-axis but can also be used for other types of rotations. The conversation also touched on transforming equations into standard form and the challenges of finding the radius and determining the appropriate differential in the integral. The teacher suggested using a different coordinate system for more complicated problems.
  • #1
flyers
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In my Cal 2 class we learned the disc method to rotate an area around an axis to obtain a volume. However we only rotated a function around either the x-axis or y axis. Let's say I take the function y=x2and y=x from the interval 0 to 1 and I want to rotate the area between these functions around the line y=x, can this be done? My teacher said something about tranforming the equations into standard form but also said we are not looking at these types of problem in my course.
 
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  • #2
The radius of your "disk" would be the line perpendicular to the axis of rotation- and that might be hard to find. Also the differential in the integral would not be "dx" or "dy" but some combination. Your teacher is probably right that the best method would be to transform the coordinate system.
 

What is the disc method for finding volume?

The disc method is a mathematical technique used to find the volume of a solid of revolution, which is a 3-dimensional object formed by rotating a 2-dimensional shape around an axis. It involves dividing the shape into infinitesimally thin discs and then calculating the volume of each disc to find the total volume of the solid.

When is the disc method used?

The disc method is commonly used in calculus to find the volume of solids with circular cross-sections, such as cylinders, cones, and spheres. It can also be used to find the volume of more complex shapes, such as toroids or spheroids.

How do you calculate the volume using the disc method?

To calculate the volume using the disc method, you first need to find the area of the base of the solid, which will be the cross-sectional area of the shape. Then, you need to integrate the area over the range of rotation to find the volume. The formula for volume using the disc method is V = π * ∫ A(x)^2 dx, where A(x) is the function that gives the area of the cross-section at a given value of x.

What are the limitations of using the disc method?

The disc method is only applicable to solids of revolution, so it cannot be used to find the volume of irregular or non-rotational shapes. Additionally, the shape must have a known axis of rotation, and the cross-sections must be circular in shape. If these conditions are not met, the disc method cannot be used to find the volume.

Can the disc method be used to find the volume of hollow objects?

Yes, the disc method can be used to find the volume of hollow objects as long as they are solids of revolution. The inner and outer radii of the object must be known in order to calculate the volume of the hollow space using the disc method.

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