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.
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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