APPLICATION OF DEFINITE INTEGRAL

This math ppt describes APPLICATION OF DEFINITE INTEGRAL . this will be useful for B.TECH students for their semester ppts.

1. B.P.PODDAR INSTITUTE OF MANAGEMENT & TECHNOLOGY CSE-A YEAR-1 ST SEM-1 ST SESSION-2019-20 APPLICATION OF DEFINITE INTEGRAL BY- 1) DEBANKA KHAN(29) 2) CHAYAN PATHAK(26) 3) DEBAJYOTI ROY CHOWDHURY(27) 4)DIGANTA GHOSH(32)

2. INDEX 1] INTRODUCTION 2]DETERMINATION OF AREA UNDER THE CURVE 3 ] FINDING VOLUME OF REVOLUTION 4] APPLICATION OF DEFINITE INTEGRAL IN VARIOUS OTHER FIELDS

3. INTRODUCTION Definite intigral is used to calculate areas between two curves, volumes, length of curves, and several other applications from real life such as calculating the work done by a force, the pressure a liquid exerts on an object , and basic statistical concept. Definite intigral is an integral expressed as the difference between the values of the integral at specified upper and lower limits of the independent variable. The major advance in integration came in the 17th century with the independent discovery of the  fundamental theorem of calculus  by  Leibniz  and  Newton .

4. Area Under the Curve How do we find areas under a curve, but above the x-axis?

5. Area Under the Curve As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate.

6. Area Under the Curve It is possible to find the exact area by letting the width of each rectangle approach zero. Doing this generates an infinite number of rectangles.

7. AN EXAMPLE OF CALCULATING AREA UNDER THE CURVE USING DEFINITE INTIGRAL Find the area between the graph of f and the x-axis on the interval [0, 1].

8. FINDING VOLUME OF REVOLUTION Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness δx , or a cylindrical shell of width δx ; and then find the limiting sum of these volumes as δx approaches 0, a value which may be found by evaluating a suitable integral.

9. FINDING THE VOLUME OF REVOLUTION USING DISC METHOD The disk method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution. The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the x-axis is given by V=  a ∫ b │ƒ(x) 2 –g(x) 2 │ dx If g(x)=0(e.g. revolving an area between the curve and the X- axis), this reduces to: V=  a ∫ b ƒ(x) 2 dx .

10. FINDING VOLUME OF REVOLUTION USING CYLINDER METHOD The cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution. The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the y-axis is given by V=2  a ∫ b x│ƒ (x) –g(x)│ dx If g(x)=0(e.g. revolving an area between the curve and the y- axis), this reduces to: V= 2 a ∫ b x│ƒ (x) │ dx .

11. APPLICATION OF DEFINITE INTEGRAL IN VARIOUS OTHER FIELDS APPLCATION IN ENGINEERING:. 1:An Architect Engineer uses integration in determining the amount of the necessary materials to construct curved shape constructions (e.g. dome over a sports arena) and also to measure the weight of that structure. Calculus is used to improve the architecture not only of buildings but also of important infrastructures such as bridges. 2:In Electrical Engineering, Calculus (Integration) is used to determine the exact length of power cable needed to connect two substations, which are miles away from each other. 3:Space flight engineers frequently use calculus when planning for long missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon.

12. 2) In determination of cardiac output APPLICATION IN MEDICAL SCIENCE: 1)Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed.

13. APPLICATION IN PHYSICS: 1) In Physics, Integration is very much needed. For example, to calculate the Centre of Mass, Centre of Gravity and Mass Moment of Inertia of a sports utility vehicle. 2)To calculate the velocity and trajectory of an object, predict the position of planets, and understand electromagnetism. Application in Statistics: 1)Statisticians use calculus to evaluate survey data to help develop business plans for different companies. Because a survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction for the appropriate action.

14. Application in Research Analysis: An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits. Application in Graphics: A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. It can create a realistic environment for movies or video games. Application in Chemistry: It is used to determine the rate of a chemical reaction and to determine some necessary information of Radioactive decay reactio