- #1
nameVoid
- 241
- 0
y=1/x, y=0, x=1,x=1, rotate around y axis;
curious how to set this up using washers
curious how to set this up using washers
nameVoid said:y=1/x, y=0, x=1,x=1, rotate around y axis;
curious how to set this up using washers
The formula for finding volume from rotating a function f(x) about the y-axis is V = π∫(f(y))^2 dy, where f(y) is the function in terms of y.
The limits for finding volume from rotating a function about the y-axis are determined by the intersection points of the function with the y-axis. These points will be used as the lower and upper limits for the integral.
The disk/washer method involves slicing the rotated shape into thin disks or washers along the y-axis, and then finding the volume of each disk or washer. The volume of each disk is π(r)^2 dy, while the volume of each washer is π(R)^2 dy, where r is the radius of the smaller disk and R is the radius of the larger disk.
The shape of the function determines the radius of the disks/washers and therefore affects the volume. If the function is a straight line, the radius remains constant and the volume is a cylinder. If the function is curved, the radius changes and the volume is a cone, cylinder, or frustum depending on the shape of the function.
Yes, for example, if we want to find the volume of the solid generated by rotating the function y = x^2 about the y-axis from x = 0 to x = 2, we would use the formula V = π∫(x^2)^2 dx from 0 to 2, which simplifies to V = π∫x^4 dx from 0 to 2. This can be solved using integration to find the volume as π(32/5) = 6.4π units cubed.