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Integrals
- Introduction of Integrals
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- Methods of Integration> Integration by Substitution
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Definition: Definite integral
A definite integral represents the value of a function accumulated between two limits.
It can also be interpreted geometrically as the net area between the graph of y = f(x) and the x-axis from x = a to x = b.
Indefinite vs definite integral
| Feature | Indefinite Integral | Definite Integral |
|---|---|---|
| Notation | \[\int f(x) \, dx\] | \[\int_{a}^{b} f(x) \, dx\] |
| Result | Function family | Numerical value |
| Constant C | Present | Not present |
| Limits | No limits | Lower and upper limits |
| Meaning | Antiderivative | Net area / accumulated value |
Example 1
Evaluate:
Solution: An antiderivative of x is \[x^2 / 2\]. So,
Answer: 1/2
Example 2
Evaluate:
Solution: An antiderivative of \[2x + 1\] is \[x^2 + x\]. Hence,
Answer: 10
Real Life Examples
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Distance from velocity: If velocity changes with time, a definite integral gives the total displacement over a time interval.
-
Economics: Total change can be obtained from a rate function, such as marginal quantity accumulated over a fixed interval.
-
Area measurement: Definite integrals help measure curved regions more accurately than ordinary geometry formulas.
Key Points: Definite Integrals
-
Used to find exact accumulated value over a fixed interval.
-
Written as \[\int_{a}^{b} f(x) \, dx\].
-
Evaluated using \[F(b) - F(a)\].
-
Gives a unique numerical value.
-
Represents net area geometrically.
