Find Volume of Revolution for y=2+x and y=x^2 about y-axis | Shell Method

In summary, the task is to find the volume generated by rotating the area bounded by y=2+x and y=x^2 about the y-axis using the shell method. There is a debate on whether to include the left part of the solid or not since it is covered by the right part. However, if the problem specifies rotating around the y-axis, then the left part should not be counted.
  • #1
miz_ai
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Homework Statement


Find the volume generated by rotating the area bounded by y=2+x and y=x^2 about the y-axis.


Homework Equations


Volume of revolution.

The Attempt at a Solution


shell method
integral (0 to 2) of x(2+x-x^2) dx

I think this can be solved by eliminating the left part and only count the volume of the left side, because the left part of the solid result is covered by the right part. It's a problem in my school because our teacher is debating whether the left part is counted or not.

i'm sorry i can't write in equotion.. I'm newbie
 
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  • #2
IF the problem really says "rotate around the y-axis, then you are right: the solid is generated is by the region between y= x+ 2 and y= x2 for x> 0. The problem would make more sense and be more interesting if it were rotated around the x-axis.
 

What is the definition of "Find Volume of Revolution"?

The "Find Volume of Revolution" refers to a mathematical process of calculating the volume of a three-dimensional object created by rotating a two-dimensional curve around a specified axis.

What is the "Shell Method"?

The Shell Method is a technique used to find the volume of a solid of revolution by slicing it into thin cylindrical shells and integrating their volumes.

How do you find the volume of revolution for y=2+x and y=x^2 about y-axis using the Shell Method?

To find the volume of revolution for y=2+x and y=x^2 about y-axis using the Shell Method, we first need to set up the integral as follows: V = ∫2πx((2+x)-(x^2))dx. Then, we integrate the equation with the limits of integration being the x-values where the two curves intersect. Finally, we multiply the integral by 2π to get the final answer.

What are the key steps involved in solving for the volume of revolution using the Shell Method?

The key steps involved in solving for the volume of revolution using the Shell Method are:

  1. Identifying the axis of rotation.
  2. Setting up the integral by expressing the volume in terms of the variable of integration.
  3. Determining the limits of integration by finding the points of intersection between the curves.
  4. Integrating the equation using the appropriate technique.
  5. Multiplying the integral by 2π to get the final answer.

What are some real-life applications of finding the volume of revolution using the Shell Method?

The Shell Method has many real-life applications in fields such as engineering, physics, and architecture. Some examples include calculating the volume of water in a cylindrical tank, finding the volume of a rotating machine part, or determining the volume of a bridge support.

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