Finding volume of arch dam

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In summary, the problem involves finding the volume of a rotation using the shell method and a given axis and rotation. The cross section is made up of three parts: two triangles and rectangles. The solution involves applying the shell method on each of these parts and integrating to find the volume. The distance to the axis of revolution remains constant at x+220 in each application of the shell method.
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Sentral
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Homework Statement



The problem is number 4 at this link: http://college.cengage.com/mathematics/larson/calculus_early/2e/students/downloads/mws6a.pdf

The axis of revolution is 150 and the rotation is 150 degrees.

Homework Equations



Using the shell method.

My offset is x+220.

The equations that make up the cross section:

[tex].03x^2+7.1x+350[/tex]

[tex]-6.593x+389[/tex]

[tex]389[/tex]

The Attempt at a Solution



I planned on splitting the cross section into three parts, which are the 2 triangles and rectangles, and apply the shell method on each of these. So first I did [tex]2pi*Integral[(x+220)(.03x^2+7.1x+350)][/tex] with the bounds being -70 to -16. Then I multiplied this answer by (150/360) since it's only 150 degree rotation. In each of my applications of the shell method, the p(x), or the distance to the axis of revolution stayed at x+220. Am I on the right track if I do this for each of the 3 sections?
 
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  • #2
Welcome to PF!

Hi Sentral! Welcome to PF! :smile:

(have a pi: π and an integral: ∫ and try using the X2 tag just above the Reply box :wink:)

I'm not sure what you're doing (what's 220? :confused:) …

the shell method involves finding the volume of the arc-shaped slice (it will be 150/360 of the volume of a complete cylinder) from x to x + dx …

it will be something times dx …

and then integrating. :smile:
 

1. How do you calculate the volume of an arch dam?

The volume of an arch dam can be calculated by multiplying the cross-sectional area of the dam by its length. The cross-sectional area can be determined by taking multiple measurements of the width and height of the dam at regular intervals and using the formula for the area of a trapezoid: A = (1/2)h(b1 + b2), where h is the height of the trapezoid and b1 and b2 are the lengths of the parallel sides. These measurements can then be integrated to find the total cross-sectional area, which can then be multiplied by the length of the dam to find the volume.

2. What units are typically used to express the volume of an arch dam?

The volume of an arch dam is typically expressed in cubic meters (m3) or cubic feet (ft3), depending on the unit system used for the measurements. It is important to use consistent units throughout the calculations to ensure accuracy.

3. How does the shape of an arch dam affect its volume?

The shape of an arch dam can greatly affect its volume. A wider arch dam will have a greater cross-sectional area and therefore a larger volume. Similarly, a taller arch dam will have a larger height and therefore a larger cross-sectional area and volume. The shape of the arch itself can also impact the volume, with a flatter or more curved arch resulting in a different cross-sectional area and volume.

4. Are there any other factors that can affect the volume of an arch dam?

Yes, there are other factors that can affect the volume of an arch dam. These include the materials used in construction, any internal voids or hollow spaces within the dam, and the angle of the arch. These factors can all impact the cross-sectional area and therefore the overall volume of the dam.

5. How is the volume of an arch dam used in its design and construction?

The volume of an arch dam is a crucial factor in its design and construction. It is used to determine the amount of materials needed for construction, as well as the structural integrity and stability of the dam. The volume also plays a role in determining the water storage capacity and potential energy production of the dam.

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