Surface area of solid of revolution

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.

\dpi{120} \mathbf{S=\int_{a}^{b}2\pi (radius)(arc length)dx}

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How to find Arc Length of different curves ?

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…

\dpi{120} \mathbf{L=\int_{a}^{b}\sqrt{1+[f'(x)]^{2}}dx}

Find more detailed work and examples here…

Fluid Pressure and Force as applications of integral.

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.

What is First order Linear Differential equation and how to solve them ?

What is Separable differential equation and how to solve them?

Get answers of all these questions here…..

Mass, Work and Energy problems as applications of Integral Calculus

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.

\dpi{120} \mathbf{M=\int_{a}^{b}\rho (x) dx}

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,

\dpi{120} \mathbf{W=\int_{a}^{b}F(x) dx}

Pumping liquids from Tanks :

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.

For more help on this topic you can visit..

Logistic Differential Equations (LDE)

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.

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Understanding Related Rates

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!

What is the difference between linear approximation and linearization.

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.

Applications of integral 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


Applications of Differential calculus

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.

Don’t forget to share your experience  and your valuable suggestions to improve  this service. Thanks !