Applications Of Integration Applications Of Integration One of the important fields of mathematics is integration. The meaning of Integration can vary according to the use. There are various fields in which it can be used like social and economy, mathematics, engineering and many more. Isaac Newton and Gottfried Leibniz in the late 17th century formulated the concepts of Integration. They independently developed the fundamental theorem of calculus. Integration is closely related to differentiation. There are many fields that utilize Application of Integration. Finding out the area under the curve was considered a difficult job in the earlier times but by using integration we can easily calculate it. There are many fields where integration is used like engineering, physics, economics, electronics and even in daily life too. A very useful application of integration is in the field of distance or displacement, velocity or speed and acceleration. We can easily find out an expression of displacement by integrating velocity and an expression of velocity by the given acceleration i.e. Displacement from Velocity and Velocity from Acceleration.
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Integration is widely used by engineers. It is well known as system integration which is successfully done in construction engineering. This level of integration can solve many big problems and can help in achieving much more than financial gains. Some applications of Integration: 1. Volume :- It is similar to calculating the area of planar regions by integration. The area of region can be calculated by dividing into pieces and adding up the area of the pieces. Once a formula is obtained for differential increment in the area we can calculate area by integration. This process can be used for volume, arc length and work. 2. Arc Length :- By calculating the length of a small portion of an arc we can integrate the length of that portion from lower limit to higher limit which will result into the length of the arc. 3. Work :- Work is a product of force and distance. It is easy to calculate when force is constant but difficult when force is a variable. For example, when we put a rocket in orbit, the force varies from time to time at that moment we have to calculate the work with the help of integration. 4. Pascal’s Principle: The pressure exerted at a depth h in a fluid is the same in every direction. If the area of a plate A then the force of the plate entirely depend upon height h. Integration can be used in measuring the voltage across a capacitor. The current in an electric circuit is equal to the time rate of change of the charge that passes to that point in the circuit. The force between charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.
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Integration is also used as social integration which is the scientific study of society is the movement of minority groups such as refugees and underprivileged sections of a society into the mainstream of societies. It helps the government to study this concept and make policies to develop the society. (3) The brick hits the ground when h (t) = 0 that is, −8 t2 + 20 = 0. Solving for t > 0 we find t ≈ 1.6 sec. At that time, v = −16 (1.6) = −25.6 m / sec. (The velocity is negative because we are decide up as positive and down as negative.) We can take another example of anti derivative as we have a function ∫ cos (3a) da=? So as the solution of this function as the derivative of the sine is cosine so anti derivative of cosine is sine. For solution of it we can take the bottom up approach that the anti derivative of the function most probably is sin (3x).So if we take the derivative sin (3x) then what do: d / da sin (3a)=cos (3a) d/ da (3a) = 3 cos (3a) So the anti derivative of it is almost sin (3x) but there is some other factor of 3 which comes at differentiation. It is because of chain rule of differentiation. Then we calculate little bit of changing function of sine as:-
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