2022年4月26日 · Suppose the torus is obtained by rotating a circular region whose centre is at (0, R) with the equation: x 2 + (y – R) 2 = r 2. Now, we know this circular region is lying between the curves, x = and y = . Using Washer method, the volume of the given solid torus can be written as, V = = = = = (4πR) (1/2) (πr 2) = 2π 2 r 2 R

2025年1月18日 · Key Idea 24: The Washer Method Let a region bounded by \(y=f(x)\), \(y=g(x)\), \(x=a\) and \(x=b\) be rotated about a horizontal axis that does not intersect the region, forming a solid. Each cross section at \(x\) will be a washer with outside radius \(R(x)\) and …

By rotating the circle around the y-axis, we generate a solid of revolution called a torus whose volume can be calculated using the washer method. Washer method. We revolve around the y-axis a thin horizontal strip of height dy and width R - r. This generates a disk with a hole in it (a washer) whose volume is dV.

美 国的微积分教材把它叫做Washer Method。 垫圈法的思路是将旋转体分成很多很薄的 形如垫圈的薄片，然后利用定积分将这些 垫圈的体积累积起来，得到旋转体的体积。

Washer Method Formula. The washer method is used to find the volume enclosed between two functions. In this method, we slice the region of revolution perpendicular to the axis of revolution. We call it as Washer Method because the slices obtained in this way forms washers.

The washer method is a way to find the volume of objects of revolution. It’s a modification of the disc method for solid objects to allow for objects with holes. It’s called the “washer method” because the cross sections look like washers. A thin, horizontal slice from the torus on the left is rotated around the y-axis.

hint: in addition to what Raskolnikov said, it may be simpler to consider the washer method instead of the shell method for setting up your integral. (That is, integrate in y y and consider the area of annuli of the constant y y slices.) And you probably know …

2014年8月30日 · By Washer Method, the volume of the solid of revolution can be expressed as: #V=pi int_{-r}^r[(sqrt{r^2-x^2}+R)^2-(-sqrt{r^2-x^2}+R)^2]dx# , which simplifies to:

Use the disk/washer method to find the volume of the general torus if the circle has radius r and its center is R units from the axis of rotation. For part a, I started by rewriting equation as x = 2 ± 1 −y2− −−−−√ x = 2 ± 1 − y 2. I was using the washer setup, and simplified to V = 8π∫1 −1 1 −y2− −−−−√ dy V = 8 π ∫ − 1 1 1 − y 2 d y.

2012年10月4日 · The washer method is just a special case of taking slices. Slice the torus perpendicular to the x-y plane, going from y=-a to y=a. Each slice is an annulus = a disk with an inner disk removed. At height y, the outer radius is the distance from (b, y) to (x_1, y), where (x_1, y) is on the original circle with x_1 <0.