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.

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