Application of Integrals Application of Integrals Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral, is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b, such that area above the x-axis adds to the total, and that below the xaxis subtracts from the total. The term integral may also refer to the notion of the antiderivative, a function F whose derivative is the given function f. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. I Know More About :- Operation Multiplication
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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. 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. Read More About :- Powers of Numbers
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