Area Between Curves: Calculate Volume Revolved Around Y Axis

In summary, the conversation revolved around finding the volume of a solid when a region is revolved around the y-axis. The individual attempted to solve the problem by graphing and using the formula A=pi*\sqrt{1+y}^2, but realized they used the wrong equation and should have set it equal to y instead of x. After correcting their mistake, they came to the solution of 8*pi for the volume of the solid.
  • #1
Aerosion
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0

Homework Statement



This is so basic and yet I'm not getting it.

Find the volume of the solid when the region enclosed is revolved arond the y axis.

[tex] x = \sqrt{1 + y}[/tex]
x=0, y=3

Homework Equations





The Attempt at a Solution



So I first graphed the thing, finding that the enclosed area was between y=1 and y=3. I took the volumne of the equation, or A=pi *[tex]\sqrt{1 + y}^2[/tex]. I then took the integral, pi * [tex]\int_{1}^{3} 1 + y dy[/tex], coming out with 6*pi. My book says this isn't right. Did I do anything wrong? Use the wrong formula?
 
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  • #2
The enclosed area is not between y=1 and y=3.
 
  • #3
Oh wait...I think I got it.

I was graphing with the equation set equal to x, when it should have been set equal to y. This would make [tex]\int_{-1}^{3} 1 + y dy[/tex], which, when multiplied by pi, would get 8*pi.

Thanks.
 

Related to Area Between Curves: Calculate Volume Revolved Around Y Axis

1. How do you calculate the volume of a shape revolved around the y-axis?

To calculate the volume of a shape revolved around the y-axis, you will need to use the formula V = π∫(R^2 - r^2)dx, where R is the outer radius and r is the inner radius. You will also need to determine the limits of integration and use the appropriate integrals to find the volume.

2. What is the difference between using the washer method and the shell method to find the volume?

The washer method is used when the cross-sections of the shape are perpendicular to the axis of rotation, while the shell method is used when the cross-sections are parallel to the axis of rotation. In the washer method, you use the formula V = π∫(R^2 - r^2)dx, while in the shell method, you use the formula V = 2π∫r(x)h(x)dx.

3. How can you determine the outer radius and inner radius for a shape revolved around the y-axis?

The outer radius is the distance from the axis of rotation to the outer boundary of the shape, while the inner radius is the distance from the axis of rotation to the inner boundary of the shape. To determine these values, you may need to use algebraic or geometric methods to find the equations for the boundaries of the shape.

4. Can you use the same method to find the volume of a shape revolved around the x-axis?

No, the method for finding the volume of a shape revolved around the x-axis is different. You will use the formula V = π∫(R^2 - r^2)dy for the washer method, and V = 2π∫r(y)h(y)dy for the shell method. You will also need to determine the limits of integration and use the appropriate integrals.

5. What are some real-world applications of finding the volume of a shape revolved around the y-axis?

The volume of a shape revolved around the y-axis is often used in engineering and architecture to calculate the volume of objects such as pipes, cylinders, and bottles. It can also be used in manufacturing to determine the volume of objects created through rotational molding or extrusion processes.

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