How to Find the Volume of a Rotating Solid?

Also, the link you provided is broken.In summary, the problem involves finding the volume of a 3D solid generated by rotating the curve f(x) = 2 sin x on the interval [0, π] around the line y = -1. The attempted solution involves using the integration of pi∫(2sin(x) + 1)^2 dx, but there seems to be confusion about the definition of the region and the boundary. The provided link is also broken.
  • #1
DrAlexMV
25
0

Homework Statement


Region: f(x) = 2 sin x on the interval [0, π]. Find the volume of
the 3D solid obtained by rotating this region
about the dashed line y = −1.

Homework Equations



Integration of pi∫(2sin(x) + 1)^2 dx

The Attempt at a Solution



http://www3.wolframalpha.com/Calculate/MSP/MSP85361a550f47a75i0g7800002h3g026g9d9ieg7f?MSPStoreType=image/gif&s=3&w=301&h=35

That does not seem to work. I am completely baffled as to what I am doing wrong.
 
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  • #2
f(x) = 2 sin x on the interval [0, π] does not define a region. It defines a curve and two limits on the x coordinate. You seem to be assuming the remaining boundary is the line y = -1, but why that rather than, say, the x axis?
 

FAQ: How to Find the Volume of a Rotating Solid?

1. What is the volume of a rotating solid?

The volume of a rotating solid is the amount of space occupied by the solid object as it rotates around an axis.

2. How is the volume of a rotating solid calculated?

The volume of a rotating solid can be calculated using the formula V = πr^2h, where r is the radius and h is the height of the solid.

3. What is the difference between the volume of a rotating solid and a regular solid?

The volume of a rotating solid takes into account the rotational motion of the object, whereas the volume of a regular solid does not consider any movement.

4. Can the volume of a rotating solid be negative?

No, the volume of a rotating solid cannot be negative as it represents a physical measurement of space and cannot have a negative value.

5. How does the shape of a rotating solid affect its volume?

The shape of a rotating solid directly impacts its volume, as different shapes have varying formulas for calculating their volume. For example, a cylinder and a cone have different volume formulas despite having the same height and radius.

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