Hello jzsc. Welcome to PF !jzsc said:Homework Statement
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
y=x, y=0, x=2, x=4; about x=1
The Attempt at a Solution
one of the radii is x=y but I'm not sure how to find the other one and the interval? Please explain this one to me. Thanks!
e^(i Pi)+1=0 said:[itex]2\Pi\int x f(x) dx[/itex]
(I don't know how to put the a & b above and below the integrand)
A solid of revolution is a three-dimensional object created by rotating a two-dimensional shape around an axis. This results in a shape that has volume, rather than just area like the original two-dimensional shape.
Commonly used shapes in solids of revolution include circles, rectangles, and triangles. However, any two-dimensional shape can be used as long as it is rotated around an axis.
The formula for calculating volume of a solid of revolution is V = π∫ab (f(x))2 dx, where a and b are the limits of the shape being rotated and f(x) is the function that defines the shape.
The axis of rotation is determined by the shape and the desired result. For example, if you want a cylinder, the axis of rotation should be perpendicular to the base of the shape. If you want a cone, the axis of rotation should be at an angle to the base of the shape.
Solids of revolution have many real-world applications, including creating objects such as bottles, vases, and cones. They are also used in engineering and architecture for designing and constructing structures with curved surfaces.