If we revolve a curve y=f(x) on the interval [a,b] about x axis, then we calculate area of resulting surface by breaking the curve into pieces as we did for arc length. A piece of curve of length dS at an average distance y from x axis traces out a surface that is well approximated by a slice of a cone whose area is approximated by 2πydS.
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Length of a curve is called arc length.The method of finding arc length for different type of curves is very much similar, yet different formulas are used for them. Here we are going to study these formulas one by one.
To find arc length of a curve defined by function f(x) over a certain interval [a,b] we use the following formula…
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Hydrostatic pressure and force is an important application of integrals, also used in Physics. In this topic we deal with pressure and force exerted by fluids on submerged plates.
When rectangular plate is submerged horizontally then force acting on it, is constant but when it is submerged vertically in water then pressure acting on it, is not constant throughout, so force acting on it must be calculated using slicing which leads to integral. Here you can find some examples on this topic. https://celestialtutors.com/subtopic/fluid-pressure-and-force/
What is Separable differential equation and how to solve them?
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Integration is used to calculate mass of a given object based on a density function. We can calculate mass of a one dimensional and two dimensional object using density function.
Work is said to be done if a force F , working on an object displaces the body through some distance dx. Let F(x) represents the force at point x, then the work done over the interval [a,b] is given as,
The method of slicing the object into small pieces and moving each piece all the way to the top applies very nicely to situations where water or any liquid is being pumped from a tank. The work integral so obtained will depend on the shape and geometry of slices that occur in each problem.
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A Logistic Differential Equation (LDE) is an ordinary differential Equation whose solution is a logistic function.
LDE(logistic differential equation) include two positive parameters –
i) Growth parameter (growth rate)k: This parameter plays a role similar to that of r in exponential differential equation.
ii) Carrying capacity (M):The carrying capacity of an organism in a given environment is defined to be the maximum population of that organism that the environment can sustain indefinitely.
This is an important application of differential calculus. Related rates is nothing but the study of relations among different variables for the ongoing situation and how they are changing with respect to another quantities. Here you can find real life word problems on related rates. Process of solving such problems is explained beautifully!
There is no difference between linear approximation and linearization. These are just two different ways to say the same thing.
Linear approximation is an important application of differential calculus.
Integral has many applications right from finding areas under the curve to finding volume of solids of revolution, finding arc length, surface area, work done etc and many other applications of physics too.
Here you can find beautiful summary of finding volume by three methods (disk, washer and shell) as well as volume by cross sections along with examples. Finding area under the curve is also discussed
Differential calculus has numerous applications , from finding critical values to finding maximum and minimum values , checking concavity, rate of change measures, linearization , optimization and many more.
I find these topics very interesting and i bet you too will find them interesting once you get the concepts. Here i’m sharing a link where you can find useful notes on these topics. Some are added already and some are being added at regular intervals.
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