This section ends with a discussion of the
theorem of Pappus for volume , which allows us to find the volume of particular kinds of solids by using the centroid. (There is also a theorem of Pappus for surface area, but it is much less useful than the theorem for volume.)
Theorem of pappus for volume
Let
R be a region in the plane and let
l be a line in the plane that does not intersect
R . Then the volume of the solid of revolution formed by revolving
R around
l is equal to the area of
R multiplied by the distance
d traveled by the centroid of
R.
Proof
We can prove the case when the region is bounded above by the graph of a function
and below by the graph of a function
over an interval
and for which the axis of revolution is the
y -axis. In this case, the area of the region is
Since the axis of rotation is the
y -axis, the distance traveled by the centroid of the region depends only on the
x -coordinate of the centroid,
which is
where
Then,
and thus
However, using the method of cylindrical shells, we have
So,
and the proof is complete.
□
Using the theorem of pappus for volume
Let
R be a circle of radius 2 centered at
Use the theorem of Pappus for volume to find the volume of the torus generated by revolving
R around the
y -axis.
The region and torus are depicted in the following figure.
The region
R is a circle of radius 2, so the area of
R is
units
2 . By the symmetry principle, the centroid of
R is the center of the circle. The centroid travels around the
y -axis in a circular path of radius 4, so the centroid travels
units. Then, the volume of the torus is
units
3 .
Let
R be a circle of radius 1 centered at
Use the theorem of Pappus for volume to find the volume of the torus generated by revolving
R around the
y -axis.
Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
For point masses distributed along a number line, the moment of the system with respect to the origin is
For point masses distributed in a plane, the moments of the system with respect to the
x - and
y -axes, respectively, are
and
respectively.
For a lamina bounded above by a function
the moments of the system with respect to the
x - and
y -axes, respectively, are
and
The
x - and
y -coordinates of the center of mass can be found by dividing the moments around the
y -axis and around the
x -axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product