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Fundamental Theorem of Calculus- 2 Video Lecture | Mathematics for Competitive Exams
FAQs on Fundamental Theorem of Calculus- 2 Video Lecture - Mathematics for Competitive Exams
| 1. What is the Fundamental Theorem of Calculus? |  |
Ans. The Fundamental Theorem of Calculus is a fundamental result in calculus that establishes the relationship between differentiation and integration. It states that if a function f is continuous on the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to the difference of the antiderivative evaluated at the endpoints: ∫[a to b] f(x) dx = F(b) - F(a).
| 2. How is the Fundamental Theorem of Calculus used in practice? | |
Ans. The Fundamental Theorem of Calculus has numerous applications in mathematics and science. It allows us to calculate definite integrals by finding the antiderivative of a function and evaluating it at the endpoints of the interval. This theorem is used to solve a wide range of problems in areas such as physics, engineering, economics, and statistics.
| 3. Can the Fundamental Theorem of Calculus be applied to any function? | |
Ans. The Fundamental Theorem of Calculus can be applied to any function that satisfies the conditions of continuity on the interval of integration. If the function is continuous, then an antiderivative exists, and the theorem can be used to evaluate the definite integral. However, for functions that are not continuous or have discontinuities within the interval, the theorem may not be applicable.
| 4. Is it possible to use the Fundamental Theorem of Calculus to find indefinite integrals? | |
Ans. No, the Fundamental Theorem of Calculus is specifically used to evaluate definite integrals. To find indefinite integrals, which represent a family of antiderivatives, other techniques like integration by substitution, integration by parts, or the use of integral tables are used.
| 5. Are there any limitations or conditions when applying the Fundamental Theorem of Calculus? | |
Ans. Yes, there are certain conditions that need to be met when applying the Fundamental Theorem of Calculus. The function being integrated must be continuous on the interval of integration, and an antiderivative of the function must exist. Additionally, the interval of integration should be finite. If any of these conditions are not satisfied, the theorem may not be applicable, and alternative methods may be needed to evaluate the integral.
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