How Do You Calculate the Volume of a Solid Rotated Around x=2?

In summary, the conversation discusses finding the area enclosed by two curves and using this information to calculate the volume of a solid when rotated around a specific line. The solution involves using integration and the formula V=PI*y^2dx. The correct range for the integral is from 0 to 1.
  • #1
joe007
23
0
Volume Of solids question help!

Homework Statement



i) find the area enclosed by the curves y=x^1/2 and y=x^4
ii)find the volume of the solid when the area in part (i) is rotated about the the line x=2

Homework Equations


V=PI*y^2dx


The Attempt at a Solution


wel the area is simple integral 0 to1 root x -x^.5 and i got 7/15

but I am not sure how to calculate the volume about x=2
 
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  • #2
hi joe007! :smile:
joe007 said:
ii)find the volume of the solid when the area in part (i) is rotated about the the line x=2

slice the volume into horizontal "washers" of height dy, and integrate :wink:
 
  • #3


Equivalent to "washers": Use "disks" to find the volume when [itex]y= x^{1/2}[/itex] is rotated around the x-axis, then use "disks" to find the volume when [itex]y= x^4[/itex] is rotated around the x-axis, and subtract.
 
  • #4


so its V=2PI* (integral 0 to 2) (x-2)x^0.5 -(x-2)x^4 dx
 
  • #5


i think it should be from 0 to 1
 
  • #6


joe007 said:
i think it should be from 0 to 1
Correct !
 

FAQ: How Do You Calculate the Volume of a Solid Rotated Around x=2?

1. What is the definition of volume of a solid?

The volume of a solid is the amount of space occupied by the solid in three-dimensional space. It is typically measured in cubic units, such as cubic centimeters (cm3) or cubic meters (m3).

2. How do you calculate the volume of a solid?

The formula for calculating the volume of a solid depends on its shape. For example, the volume of a cube is calculated by multiplying the length, width, and height of the cube (V = l x w x h). The volume of a cylinder is calculated by multiplying the area of the base by the height of the cylinder (V = πr2h).

3. Can the volume of a solid change?

Yes, the volume of a solid can change if its dimensions change. For example, if you cut a cube in half, the volume of each half will be half of the original volume. Additionally, the volume of a solid can also change if its temperature or pressure changes.

4. What is the difference between volume and surface area of a solid?

Volume and surface area are both measurements of a solid, but they represent different properties. Volume measures the amount of space occupied by a solid, while surface area measures the total area of the solid's surfaces. In other words, volume is a measurement of the solid's size, while surface area is a measurement of its outer covering.

5. How is the volume of a solid used in real life?

The concept of volume is used in many real-life applications, such as calculating the capacity of a container, determining the amount of material needed to fill a space, or measuring the amount of liquid in a container. It is also used in many scientific fields, such as chemistry, physics, and engineering, to understand and solve problems related to the physical properties of solids.

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