The document introduces integrals and their applications. It states that integrals can be used to find areas, volumes, and average values in fields like physics, engineering, economics, and geometry. It then lists some key applications of integrals, including calculating areas under curves, finding volumes of solids by rotating curves, solving motion and accumulation problems in physics, calculating probabilities and expected values, analyzing economic variables, and solving differential equations.

1. WELCOME TO
CLASS XII
APPLICATION OF INTEGRALS

2. INTRODUCTION
Integrals have applications in finding areas,
volumes, and average values. They are used in
fields such as physics, engineering, economics,
and geometry. Integrals help solve real-world
problems by providing a framework for analyzing
complex systems.

3. AREA UNDER SIMPLE CURVES

5. APPLICATIONS OF INTEGRALS
1. AREA CALCULATION: ONE OF THE MOST COMMON APPLICATIONS OF
INTEGRALS IS FINDING THE AREA UNDER A CURVE. BY CALCULATING
THE DEFINITE INTEGRAL OF A FUNCTION OVER A SPECIFIC INTERVAL,
WE CAN DETERMINE THE AREA ENCLOSED BY THE CURVE AND THE X-
AXIS.
2.VOLUME CALCULATION: INTEGRALS ARE ALSO USED TO CALCULATE
VOLUMES OF THREE-DIMENSIONAL SHAPES. BY ROTATING A CURVE
AROUND A SPECIFIC AXIS AND INTEGRATING THE AREAS OF THE
RESULTING CROSS-SECTIONS, WE CAN FIND THE VOLUME OF THE SOLID
FORMED.

6. 3.MOTION AND ACCUMULATION PROBLEMS: INTEGRALS ARE ESSENTIAL
IN PHYSICS AND ENGINEERING FOR SOLVING MOTION AND
ACCUMULATION PROBLEMS. THE INTEGRAL OF A VELOCITY FUNCTION
GIVES US THE DISPLACEMENT OF AN OBJECT OVER A SPECIFIC TIME
INTERVAL, WHILE THE INTEGRAL OF AN ACCUMULATION FUNCTION GIVES
US THE TOTAL AMOUNT OF SOMETHING ACCUMULATED OVER TIME.
4. PROBABILITY AND STATISTICS: INTEGRALS ARE USED
IN PROBABILITY THEORY AND STATISTICS TO CALCULATE
PROBABILITIES AND EXPECTED VALUES. THE AREA UNDER A
PROBABILITY DENSITY FUNCTION CURVE REPRESENTS THE PROBABILITY
OF A SPECIFIC EVENT OCCURRING.

7. 5.ECONOMIC ANALYSIS: IN ECONOMICS, INTEGRALS ARE EMPLOYED
TO ANALYZE SUPPLY AND DEMAND FUNCTIONS, CALCULATE TOTAL
REVENUE AND COST, AND DETERMINE OPTIMAL PRODUCTION LEVELS.
INTEGRALS HELP IN UNDERSTANDING THE BEHAVIOR OF ECONOMIC
VARIABLES AND MAKING INFORMED DECISIONS.
6.DIFFERENTIAL EQUATIONS: INTEGRALS ARE ESSENTIAL IN
SOLVING DIFFERENTIAL EQUATIONS, WHICH ARE EQUATIONS
INVOLVING DERIVATIVES. BY INTEGRATING BOTH SIDES OF A
DIFFERENTIAL EQUATION, WE CAN FIND THE GENERAL SOLUTION AND
SOLVE VARIOUS PHYSICAL AND MATHEMATICAL PROBLEMS.