Finding Volume Using Integration: Revolving Functions Around a Line

In summary, the conversation discusses finding the volume of a solid formed by revolving two functions around a given line. The equations and attempt at a solution are provided, with a question about the accuracy of the result. The correct calculation involves finding the distance between the functions and the given line.
  • #1
EngnrMatt
34
0

Homework Statement



Find the volume of the solid formed by revolving the functions: y=x2 and y=1 about the line y=6

Homework Equations



∏∫(ro2-ri2)dx

The Attempt at a Solution



I found the outer radius to be (1-x2)-6 and my inner radius to be -6. Also, the limits of integration were -1 and 1. Plugging it all in got -224/15. This is obviously wrong because it is negative, and the positive value didn't work either. Does anyone have a clue on how to do this?
 
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  • #2
EngnrMatt said:

Homework Statement



Find the volume of the solid formed by revolving the functions: y=x2 and y=1 about the line y=6

Homework Equations



∏∫(ro2-ri2)dx

The Attempt at a Solution



I found the outer radius to be (1-x2)-6 and my inner radius to be -6. Also, the limits of integration were -1 and 1. Plugging it all in got -224/15. This is obviously wrong because it is negative, and the positive value didn't work either. Does anyone have a clue on how to do this?

Isn't the inner radius the distance from y=1 to y=6? Isn't that 5? What's the outer radius?
 
  • #3
Yes, you are right. I see now, the outer radius will be the distance between y=6 and the lower function and the inner will be the distance between y-6 and the upper function. It worked by the way, thanks.
 

FAQ: Finding Volume Using Integration: Revolving Functions Around a Line

1. What is the concept of volume using integration?

The concept of volume using integration is a mathematical approach to finding the volume of a three-dimensional object. It involves dividing the object into infinitesimally small slices and using the integral calculus to sum up the volumes of these slices to find the overall volume of the object.

2. What are the applications of volumes using integration?

Volumes using integration has various applications in fields such as physics, engineering, and economics. It can be used to determine the volume of irregularly shaped objects, calculate the mass of an object given its density, and find the area under a curve in economics.

3. What is the formula for finding the volume using integration?

The formula for finding the volume using integration is V = ∫A(x)dx, where V represents the volume, ∫ represents the integral symbol, A(x) represents the area of the cross-section at a particular value of x, and dx represents the infinitesimal slice of thickness dx.

4. How is the volume of an object with curved sides calculated using integration?

The volume of an object with curved sides can be calculated using integration by dividing the object into infinitesimally thin slices perpendicular to the axis of rotation. The cross-sectional area of each slice can then be determined using geometry or calculus, and the volumes of these slices can be summed up using integration to find the overall volume of the object.

5. Can volumes using integration be used for objects with holes or voids?

Yes, volumes using integration can be used for objects with holes or voids. The volume of the hole or void can be subtracted from the overall volume of the object to get the final volume. This is known as the method of shells or the method of discs, depending on the shape of the object.

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