When it comes to calculus, the Shell Method and Washer Method are two techniques used to calculate the volume of a solid of revolution, a shape created by rotating a 2D figure around an axis. While they have different approaches, both methods are designed to achieve the same results. In this article, we will discuss Shell Method Vs Washer Method, their differences, and applications.
What is the Shell Method?
The Shell Method is a technique used in calculus to calculate the volume of a solid of revolution. It involves finding the volume of a shell formed by slicing the solid into thin cylindrical shells, each having a height of “dx” and a radius “r”. The volume of each shell is then calculated using the formula “2πrh*dx,” where “r” is the radius of the shell and “h” is the height of the shell. The total volume of the solid is found by adding the volumes of all the individual shells together.
Applications
The Shell Method is often used to calculate the volume of solids of revolution that have easy-to-determine functions. For instance, a solid created by revolving the curve y = √x around the x-axis from x = 0 to x = 4 can be easily calculated using the Shell Method.
What is the Washer Method?
The Washer Method is another technique used to calculate the volume of a solid of revolution. Unlike the Shell Method, the Washer Method slices the solid into discs instead of shells. A disc is created by slicing the solid perpendicular to the axis of rotation, generating a “washer” shape, just like the shape that’s formed when the washer is removed from a bolt. Each disc has an outer radius “R” and an inner radius “r,” and its volume is calculated as π(R^2-r^2)dx.
Applications
The Washer Method is often used to calculate the volume of solids of revolution with more challenging functions. For instance, a solid created by revolving the curve y = 9 – x^2 around the x-axis from x = -3 to x = 3 may be more challenging to calculate using the Shell Method, but it can be readily determined using the Washer Method.
Differences Between the Shell Method and Washer Method
The primary difference between the Shell Method and the Washer Method is the shape into which they slice the solid of revolution. The Shell Method slices the solid into thin cylindrical shells, while the Washer Method splits it into thinner discs. While the Shell Method is usually easier to use when the shape of the solid of revolution is easily identifiable, the Washer Method is often preferred when the solid has a more complicated shape.
Another difference between these techniques is the “dx” height. In the Shell Method, this height is parallel to the axis of revolution. In contrast, the “dx” height in the Washer Method is perpendicular to the axis of rotation. As a result, the “dx” values used in both techniques have different meanings.
Conclusion
In summary, the Shell Method and Washer Method are essential techniques used in calculus to calculate the volume of solids of revolution. They are two different approaches that can be employed depending on the shape and complexity of the solid of revolution. The Shell Method splits the solid into cylindrical shells and calculates the volume by adding the volumes of all the individual shells together. The Washer Method, on the other hand, slices the solid into many thin discs and determines the volume by adding the volumes of all the individual slices together. By understanding these techniques, you can choose the best method for the job and solve problems with ease.