Activity 16. Assignment
applications of calc III
Alexandra Paredes
Activity 16. Assignment
applications of calc III
Alexandra Paredes
Multiple Integral Applications in Environmental Engineering
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Calculus III Application: Finding Work Done in Forming a Right Circular Cone Mountain, Ejercicios de Cálculo
A problem-solving exercise in calculus III, where the goal is to find the work done in forming a right circular cone mountain using definite integrals. The exercise involves applying the concept of work done against gravity in calculus and involves the use of cylindrical and spherical coordinates. The problem is based on the example of Mount Fuji in Japan.
Qué aprenderás
- How is the work done in forming a mountain calculated using cylindrical coordinates?
- What is the mass of the atmosphere between the ground and an altitude of 5 km using the given density model?
- What is the work done in forming a right circular cone mountain using definite integrals?
Tipo: Ejercicios
2020/2021
1 / 8
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Activity 16. Assignment
applications of calc III
Alexandra Paredes
Activity 16. Assignment
applications of calc III
Alexandra Paredes
Multiple Integral Applications in Environmental Engineering
ApplicationApplication ExcersiceExcersice
A When studying the formation of mountain
ranges, geologists estimate the amount of work
required to lift a mountain from sea level.
Consider a mountain that is essentially in the
shape of a right circular cone. Suppose that the
weight density of the material in the vicinity of
a point is g(P) and the height is h(P).
a) Find a definite integral that represents the
total work done in forming the mountain.
B) Assume that Mount Fuji in Japan is in the
shape of a right circular cone with radius
62,000 ft, height 12,400 ft, and density a
constant 200 lb/ft^3. How much work was
done in forming Mount Fuji if the land was
initially at sea level?
A When studying the formation of mountain
ranges, geologists estimate the amount of work
required to lift a mountain from sea level.
Consider a mountain that is essentially in the
shape of a right circular cone. Suppose that the
weight density of the material in the vicinity of
a point is g(P) and the height is h(P).
a) Find a definite integral that represents the
total work done in forming the mountain.
B) Assume that Mount Fuji in Japan is in the
shape of a right circular cone with radius
62,000 ft, height 12,400 ft, and density a
constant 200 lb/ft^3. How much work was
done in forming Mount Fuji if the land was
initially at sea level?
H-z
r
R
Z
SolutionSolution
Using the integration formula in cylindrical coordinates
lb
Using the integration formula in cylindrical coordinates
lb
A model for the density of the earth’s atmosphere
near its surface is
where (the distance from the center of the earth) is
measured in meters and is measured in kilograms
per cubic meter.
If we take the surface of the earth to be a sphere
with radius 6370 km, then this model is a
reasonable one for. Use this model to estimate the
mass of the atmosphere between the ground and
an altitude of 5 km.
A model for the density of the earth’s atmosphere
near its surface is
where (the distance from the center of the earth) is
measured in meters and is measured in kilograms
per cubic meter.
If we take the surface of the earth to be a sphere
with radius 6370 km, then this model is a
reasonable one for. Use this model to estimate the
mass of the atmosphere between the ground and
an altitude of 5 km.
Application Application ExcersiceExcersice
SolutionSolution
Replacement Replacement
ReferencesReferences
Stewart James. Cálculo de varias variables. Trascendentes
tempranas. Séptima edición. ISBN: 978-607-481-898-7.
Stewart James. Cálculo de varias variables. Trascendentes
tempranas. Séptima edición. ISBN: 978-607-481-898-7.