The above is incorrect. The upper limit of integration is not 2, and your formula for the area of the typical area element is also wrong. How did you get y/4?whatlifeforme said:Homework Statement
Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and the lines about the x-axis.
Homework Equations
Y=|x|/3 , Y=1
The Attempt at a Solution
1. 2∏∫(0 to 2) (y) (Y/4 - (-Y/4)) dy
Using the correct setup, I get a volume of 4π as well.whatlifeforme said:2. y^3/6] (0 to 2)
Answer: 8∏/3 (which is wrong). the correct answer is 4∏ but I'm not sure how they go there.
Better, but still not right. If y = (1/3)|x|, then |x| = 3y, not y/3.whatlifeforme said:i also tried integrating just the right side and multiplying by two.
2* 2∏∫(0 to 1) (y) (Y/3) dy
No. It's 2x, which is not equal to 2y/3.whatlifeforme said:which i get the answer of 4pi/9.
1.if i draw a section parallel to the axis of revolutoin. the distance (radius) is y. thus the first part of the equation is right.
2.the height of the section would be the distance from (-y/3) on the left to (y/3).
You will if you use the correct value for the width of your shell.whatlifeforme said:however, if i take just the right section of the absolute value function (y/3), and integrate, and multiply by two (2). then i should have the right value.
The Shell Method is a mathematical technique used to find the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells and integrating their volume using the formula V = 2π∫(radius)(height) dx.
Yes, the Shell Method can be used to find volume from 0 to 2 as long as the solid of revolution is defined in terms of a function that can be integrated over the given interval.
The main difference between the Shell Method and the Disk Method is the shape of the slices used to calculate volume. The Shell Method uses cylindrical shells, while the Disk Method uses circular disks. In some cases, one method may be easier to use than the other depending on the shape of the solid.
Yes, there are limitations to using the Shell Method. It can only be used to find the volume of solids of revolution, meaning that the shape must be formed by rotating a curve around an axis. Additionally, the Shell Method may be difficult to use for solids with irregular shapes.
Yes, the Shell Method can be extended to find volume in higher dimensions. In three dimensions, it is known as the Cylindrical Shell Method and involves integrating over a region defined by two functions in terms of two variables.