The area under a curve is defined to be the limit of the sum of the areas of the partitioned rectangles of equal width Δx.

A definite integral is defined to be limit of such sum and hence the area under a curve in the interval under consideration is

A = $\lim_{n \to \infty }\sum_{i=1}^{n}f(x_{i})\bigtriangleup x$ = $\int_{a}^{b}f(x)dx$Based on this concept, the definite integrals are used in computing areas , volumes and also used in many Physical formulas to determine the work done, center of gravity etc.

Some of the formulas used applying integration are given below with brief notes.

**Area between two curves:**

The area between two curves f(x) and g(x) for the interval [a, b] where f(x) ≥ g(x) on [a, b] is given by

A = $\int_{a}^{b}[f(x)-g(x)]dx$

**Volume of solid of revolution:**

The volume of a solid generated by revolving y = f(x) about x axis on [a, b], using

**method of disks**V = $\int_{a}^{b}\pi y^{2}dx$

Same way the volume of the solid formed by revolving x = g(y) about y axis for a ≤ y ≤ b, using

**method****of disks** is given by

V = $\int_{a}^{b}\pi x^{2}dy$

The formulas for computing the volume of solid of revolution using the **cylindrical shell method** are as follows:

V = $\int_{a}^{b}2\pi xf(x)dx$ where 0 ≤ a < b

For computing the volume of solid obtained by rotating

about the y axis, the region under the curve f(x) from

a to b

V = $\int_{a}^{b}2\pi yg(y)dy$ The corresponding formula when the axis of rotation

happens to be the y- axis.**Applications in Physics:**

The work done by a force represented by the function f(x) in moving an object from a to b is given by

W = $\int_{a}^{b}f(x)dx$

Velocity function is obtained by integrating the acceleration and the definite integral involving the velocity

function is used in calculating the distance or displacement of a moving object.

Integration formulas are also used to compute the moment, mass and center of mass using the density function.

Integration also finds application in Economics. For example, the cost function can be retrieved from marginal cost using integration.