Calculus II Problem: Dams and intergration by slicing

In summary, the Deligne Dam on the Cayley River has a wall facing the water shaped by the region above the curve y=0.6x^2 and below the line y=164. The water level is at the top of the dam and the force exerted by water pressure can be found by multiplying the mass to be lifted by 9.8 m/sec^2. The width of the dam can be expressed as a function of y and can be found by calculating the volume and density of the water. Another method is to find the arclength of the dam.
  • #1
JasonJo
429
2
The Deligne Dam on the Cayley River is built so that the wall facing the water is shaped like the region above the curve y=0.6 x^2 and below the line y= 164 . (Here, distances are measured in meters.) The water level can be assumed to be at the top of the dam. Find the force (in Newtons) exerted on the dam by water pressure. Water has a density of 1000 kg/m^3 . Since this is a metric problem, you must multiply the mass to be lifted by 9.8 m/sec^2 to convert to a weight.
First give the integrand expressed in terms of y (the width of the dam must be expressed as a function of y).
 
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  • #2
anyone?
 
  • #3
i would find the volume of the damn using [tex]D=\frac{M}{V}[/tex]
I'm not sure on that. Once you get volume,
do volume times density to get mass. Multiply mass by 9.8 m/s^2. I think that's it, but could be wrong.
 
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  • #4
you also might be able to get the width of the dam by finding the arclength of the dam.
 

What is the purpose of using integration by slicing in calculus?

Integration by slicing is a technique used in calculus to solve problems involving three-dimensional shapes. It allows us to find the volume of irregular shapes, such as dams, by slicing them into smaller, more manageable shapes and then summing their volumes using integration.

Why is it important to understand integrals and derivatives in calculus?

Integrals and derivatives are fundamental concepts in calculus that are used to find the rate of change of a function, as well as the area under a curve. These concepts are essential in many fields, including physics, engineering, economics, and more, making it crucial to understand them for problem-solving.

How does integration by slicing differ from other integration techniques?

Integration by slicing is a specific technique used for finding the volume of three-dimensional shapes. It differs from other integration techniques, such as the fundamental theorem of calculus or integration by substitution, which are used for finding areas under curves or solving other types of integration problems.

What are some real-world applications of integration by slicing?

Integration by slicing has many practical applications, including calculating the volume of irregularly shaped objects like dams, determining the mass of an object with varying density, and finding the center of mass of a three-dimensional shape. It is also used in fields like architecture, physics, and economics.

What are some common challenges when using integration by slicing?

One of the main challenges of using integration by slicing is determining the correct slicing method and limits of integration. It can also be challenging to visualize and set up the integral for complex shapes. Additionally, numerical integration may be necessary to solve certain problems that cannot be solved analytically.

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