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Properties of Definite Integrals: Part- 1 Video Lecture | Mathematics (Maths) for JEE Main & Advanced

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FAQs on Properties of Definite Integrals: Part- 1 Video Lecture - Mathematics (Maths) for JEE Main & Advanced

1. What is a definite integral?
Ans. A definite integral is a mathematical concept used to calculate the signed area between a function and the x-axis over a specific interval. It represents the accumulation of infinitesimal areas under the curve of a function.
2. How is the definite integral denoted?
Ans. The definite integral is denoted using the integral symbol (∫), followed by the function to be integrated, the differential symbol (dx), and the interval over which the integration is performed. For example, ∫f(x)dx represents the definite integral of f(x) over the interval [a, b].
3. What are the properties of definite integrals?
Ans. The properties of definite integrals include linearity, additivity, and the property of reversing the limits. Linearity states that the integral of a sum of functions is equal to the sum of their integrals. Additivity states that the integral over the union of two intervals is equal to the sum of the integrals over each interval. The property of reversing the limits states that reversing the limits of integration changes the sign of the integral.
4. How can the definite integral be interpreted geometrically?
Ans. Geometrically, the definite integral represents the signed area between the curve of a function and the x-axis over a specific interval. If the function is positive, the integral represents the area above the x-axis, while if the function is negative, the integral represents the area below the x-axis. The value of the definite integral gives the net area between the curve and the x-axis.
5. What is the relationship between the definite integral and antiderivative?
Ans. The Fundamental Theorem of Calculus establishes the relationship between the definite integral and antiderivative. It states that if a function F is an antiderivative of a function f, then the definite integral of f from a to b is equal to the difference between F(b) and F(a). In other words, the definite integral can be evaluated by finding the antiderivative of the function and evaluating it at the upper and lower limits of integration.
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