Finding Volume Using Shell Method: x=3y-y^2

In summary, to find the volume using the shell method for the given curves x=3y-y^2 and the y-axis about the x-axis, you can find the vertex of the parabola by completing the square in the y terms and use the formula V=2pi Integral y f(y) dy, replacing x with 3y-y^2. The limits of integration can be obtained from the graph.
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Homework Statement



Have to find the volume using the shell method using the given curves

x=3y-y^2 and the y-axis about the x-axis

Homework Equations


I know to use the equation V=2pi Integral y f(y) dy but no idea where to get the high and low limits for the integral


The Attempt at a Solution

 
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Have you graphed the equation x = 3y - y^2? Its graph is a paraboloa that opens to the left. You can find the vertex of the parabola by completing the square in the y terms.

Your typical volume element is [itex]\Delta V = 2\pi*y*x*\Delta y[/itex]. Since you will be integrating with respect to y, you need to replace x in this formula with 3y - y^2. From the graph you can also get the limits of integration.
 

Related to Finding Volume Using Shell Method: x=3y-y^2

What is the shell method?

The shell method is a mathematical technique used to find the volume of a solid of revolution, where the solid is created by rotating a 2-dimensional shape around a given axis.

What is the equation for finding volume using the shell method?

The equation for finding volume using the shell method is V = 2π∫abx·f(x) dx, where a and b are the limits of integration, x is the variable of integration, and f(x) is the function representing the cross-sectional area of the solid at a given value of x.

How do you find the limits of integration for the shell method?

To find the limits of integration, you must first determine the range of values for the variable of integration (in this case, x) by solving the equation x = 3y - y^2 for y. Then, you can use these values to determine the limits of integration for x.

What is the difference between the shell method and the disk method?

The shell method and the disk method are both techniques used to find the volume of solids of revolution. However, the main difference is that the shell method integrates along the axis of revolution, while the disk method integrates perpendicular to the axis of revolution.

What are some real-life applications of the shell method?

The shell method has many real-life applications, such as finding the volume of a wine barrel or a water tower, determining the amount of paint needed to coat a cylindrical object, or calculating the volume of a human organ based on medical imaging data.

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