Using the washer method, the volume would be: V = π∫ ((x2)2- (x1)2)dy,rock.freak667 said:Revolving around the y-axis would give:
V= π∫ x2 dy
so if y=x3, then x = y 1/3 and hence x2 = y2/3
John_Smith01 said:That's what I tried before but kept getting the answer wrong. I think my textbook is wrong, did you get 1/10pi u^3?
SammyS said:Using the washer method, the volume would be: V = π∫ ((x2)2- (x1)2)dy,
where x2=y(1/3) and x1 = y(1/2).
When OP mentioned "subtracting two areas from each other", I thought he literally meant areas, not volumes. Oh well !rock.freak667 said:...
It will end up as the same since OP will have to subtract one from the other while that way will make into one simple integral.
The term "volumes of solids of revolutions" refers to the calculation of the volume of a 3-dimensional shape created by rotating a 2-dimensional shape around an axis. This is also known as the method of "disk integration" in calculus.
Any 2-dimensional shape can be used to create a volume of solid of revolution, as long as it is rotated around an axis. Some common shapes used include circles, rectangles, and triangles.
The volume is calculated by using the formula: V = π ∫ba (f(x))2dx, where π represents pi, a and b are the limits of the integral, and f(x) is the function of the rotated shape.
Volumes of solids of revolutions have many practical applications, such as calculating the volume of a water tank, the volume of a cylindrical container, and the volume of a cone-shaped ice cream cone. They are also used in engineering and architecture to calculate the volume of structures such as bridges, tunnels, and pipes.
While this method is useful for calculating volumes of certain shapes, it does have some limitations. It cannot be used for shapes that have holes or concave regions, and it is not accurate for highly irregular shapes. In addition, it can be a more complex and time-consuming method compared to other volume calculation techniques.