It is correct, but you need to include the boundaries of integration. What are they?dan38 said:Homework Statement
Just need to verify that my working is correct ^^
Need to find the volume given by the region of the xy-plane that is bounded by the curves
x = 0 and x = y − y2 .
Rotated about y-axis
Homework Equations
The Attempt at a Solution
I used the disk method.
V = pi * ∫ (y-y^2)^2 dy
Since outer radius is the curve and inner is y = 0
The Washer Method is a mathematical technique used to find the volume of a solid that is created by rotating a two-dimensional shape around an axis. It involves subtracting the volume of the hole from the volume of the larger shape using integration.
The Washer Method is specifically used when the shape being rotated has a hole in the middle, such as a washer or a ring. Other volume formulas, such as the Disk Method or the Shell Method, are used for different shapes.
The integral for the Washer Method is set up by finding the area of the larger shape and subtracting the area of the hole. This can be done by using the formula for the area of a circle (πr^2) and integrating over the desired bounds.
Yes, the Washer Method can be used for both solids of revolution and cross sections. When using it for solids of revolution, the axis of rotation is typically the x or y-axis. When using it for cross sections, the axis of rotation can be any line parallel to the x or y-axis.
One limitation of the Washer Method is that it can only be used for solids with a circular hole in the middle. If the shape being rotated has a non-circular hole or multiple holes, the Washer Method cannot be used and a different approach must be taken.