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Fundamental theorem of integral calculus (Part - 38) - Math, Class 12 Video Lecture

FAQs on Fundamental theorem of integral calculus (Part - 38) - Math, Class 12 Video Lecture

1. What is the fundamental theorem of integral calculus?
Ans. The fundamental theorem of integral calculus states that if a function f(x) is continuous on the interval [a, b] and F(x) is a function whose derivative is f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a).
2. How does the fundamental theorem of integral calculus relate to differentiation and integration?
Ans. The fundamental theorem of integral calculus connects differentiation and integration by stating that the definite integral of a function can be evaluated by finding an antiderivative of the function and subtracting the values at the endpoints of the interval. It provides a way to calculate areas under curves and also allows us to find antiderivatives of functions.
3. What is the significance of the fundamental theorem of integral calculus in real-world applications?
Ans. The fundamental theorem of integral calculus is highly significant in real-world applications as it allows us to calculate the total change or accumulation of a quantity over an interval. It is used in various fields such as physics, economics, engineering, and finance to determine quantities like velocity, displacement, area, work, and profit.
4. How does the fundamental theorem of integral calculus help in finding the area under a curve?
Ans. The fundamental theorem of integral calculus helps in finding the area under a curve by relating the process of finding the area to the calculation of definite integrals. By evaluating the definite integral of a function over a given interval, we can determine the exact area enclosed by the curve and the x-axis within that interval.
5. Can the fundamental theorem of integral calculus be applied to functions that are not continuous?
Ans. No, the fundamental theorem of integral calculus can only be applied to functions that are continuous on the given interval. If a function is not continuous, the theorem does not hold, and alternative methods such as numerical integration or approximations need to be used to find the area or evaluate the integral.
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