Find the volume of the region

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In summary, to find the volume of the solid generated by the revolution of the region enclosed by y=x^2+2, y=2x+1 and the y-axis about the line y=3, the disk/washer method or shell method can be used. The volume element can be expressed as \Delta V = \pi [(2x + 1 - 3)^2 - (x^2 + 2 - 3)^2]\Delta x. By simplifying and integrating between 0 and 1, the volume can be calculated.
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Homework Statement


Find the volume of the solid generated by the revolution of the region enclosed by y=x^2+2, y=2x+1 and the y-axis about the line y=3



Homework Equations


Disk/Washer method, shell method


The Attempt at a Solution


This one is stumping me no matter how I go about it. What I have done so far is use the disk/washer method by inputting x+3 in for f(x) and x^2-2x+1 (obtained from subtracting the two functions given) in for g(x):
V=pi[tex]\int[/tex]((x+3)^2-(x^2-2x+1)^2)dx over [0,1]. I think that because of the location of the region on a graph, however, the x+3 might become x+2, and that I might have made a mistake in the way I set the formula up, but I do not know if/where I made this mistake. If anyone can point me in the right direction and point out any mistakes I made in setting up this question it would be greatly appreciated, thanks in advance.
 
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  • #2
What you want for your typical volume element is this:
[tex]\Delta V = \pi [(2x + 1 - 3)^2 - (x^2 + 2 - 3)^2]\Delta x[/tex]
Simplify that a bit, and integrate between 0 and 1, and you should be good.
 

1. What is the definition of volume?

The volume of a region refers to the amount of space that it occupies in three-dimensional space.

2. How is volume different from area and perimeter?

Volume is a measure of the amount of space an object takes up in three dimensions, while area is a measure of the amount of surface a two-dimensional shape covers. Perimeter is a measure of the distance around the outer edge of a two-dimensional shape.

3. What is the formula for finding the volume of a region?

The formula for finding the volume of a region will depend on the shape of the region. For example, the formula for finding the volume of a cube is V = s^3, where s is the length of one side of the cube.

4. Can the volume of a region be negative?

No, the volume of a region cannot be negative. Volume is always a positive quantity because it represents the amount of space something takes up.

5. How do you find the volume of an irregularly shaped region?

To find the volume of an irregularly shaped region, you can use the method of integration. This involves dividing the region into smaller, known shapes and using calculus to calculate the volume of each shape, then adding them together to find the total volume of the region.

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