Applications of Integration Work

Integration is applied on a given function within certain limits to get various output.

Here let us see three areas of integration application work. In the normal sense, the integrations are applied to get the area under the given curve between certain limits.

1. Area under the curve f (x) between the limits ‘a’ and ‘b’ is given by`int_a^bf(x)dx`

2. The arc length of a curve f (x) between the limits ‘a’ and ‘b’ is given by `int_a^bsqrt[ 1 + [dy/dx] ^2] dx` .

3. The volume of a curve f(x) between the limits ‘a’ and ‘b’ is given by `pi int_ a^b y ^2 dx`

Now let us see problems on this topic.


Example Problems on Application of Integration Work.

Ex 1: Find the area under the curve f (x) = 2

4x 2 + 3x + 2 between 2 and 4.

Soln: Given: f (x) = 4x 2 + 3x + 2

The limits are ‘2’ and ‘4’.

Therefore Area =` int_2^4(4x ^2 + 3x + 2) dx` = `[4x ^3 / 3 + 3x ^2 / 2 + 2x] ` , the limit is between 2 to 4.

= `[(4 (4 ^3) / 3 + 3 (4) ^2 / 2 + 2 (4) ) ** (4 (2) ^3 / 3 + 3 (2) ^2 / 2 + 2 (2) )]`

Therefore Area = `96 2 / 3 ` sq units.

Ex 2: Find the arc length of a curve f (x) = 4x + 3 between 0 and 1.

Soln: Given: f (x) = 4x + 3

Therefore f ‘ (x) = 4.

Therefore arc length =` int_0^1 sqrt [(4) ^2 + 1] dx ` = `sqrt 17` `int_0^11 dx`

= sqrt 17 (x) between 0 to 1 = sqrt 17 (1 – 0) = sqrt 17 units.


More Example Problem on Application of Integration Work.

Ex 3: Find the volume generated when the finite region enclosed by the curve y = x ^2 and the lines x = 2, x = 4 and the x axis.

Soln: Volume = `pi int_2^4 y ^2 dx`

=` pi int_2^4 (x ^2) ^2 dx`

= `pi` [`x ^5 / 5` ] 2 to 4

= `pi / 5 [ 4 ^5** 2 ^5] = 992 / 5` cubic units.