Fundamental therom of calculus

In summary, the conversation is about a confusion regarding the fundamental theorem of calculus. The person is asking for an explanation on why the outcome of the equation F(a) - F(b) can be negative when using the theorem, but only gives a positive outcome when using a calculator. They also mention using the FTC and getting a different result. The other person corrects them, saying that F is not the derivative of the function, but the anti-derivative. The conversation ends with a question about the anti-derivative of sin(x).
  • #1
kmikias
73
0
Hi.guys i have some confusion on the fundamental therom of calculus.
here is my question.

1. somethimes when i use the fundamental therom of calculus which is F(a) - F(b).
the out come will be negative .in the same question when i type in my calculator its only give me positive i guess is the area. for example Integral of sinx dx from 0 to 1.5
when i did by hand using FTC its gives me -0.929 but in the calculator its 0.929.I am guesing it the calculator only give the area.(not sure)

so i need explanation on this .

F=derivative of function.
 
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  • #2


Are you sure you're not getting the endpoints around the wrong way in the calculator? On some calculators, the order in which one should put the endpoints is rather baffling,.
 
  • #3


kmikias said:
Hi.guys i have some confusion on the fundamental therom of calculus.
here is my question.

1. somethimes when i use the fundamental therom of calculus which is F(a) - F(b).
the out come will be negative .in the same question when i type in my calculator its only give me positive i guess is the area. for example Integral of sinx dx from 0 to 1.5
when i did by hand using FTC its gives me -0.929 but in the calculator its 0.929.I am guesing it the calculator only give the area.(not sure)

so i need explanation on this .

F=derivative of function.
No, in the fundamental theorem of calculus F is NOT the derivative of the function, it is the anti-derivative of the function. Now, what is the anti-derivative of sin(x)?
 

1. What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that relates the concept of differentiation with integration. It states that if a function f is continuous on an interval [a, b] and F is the antiderivative of f, then the definite integral of f from a to b is equal to F(b) - F(a).

2. Why is the Fundamental Theorem of Calculus important?

The Fundamental Theorem of Calculus is important because it allows us to solve problems involving area and accumulation. It provides a way to evaluate integrals without having to use Riemann sums, making it a more efficient method for solving complex problems in calculus.

3. Is it possible to use the Fundamental Theorem of Calculus to find the derivative of a function?

Yes, the Fundamental Theorem of Calculus has two parts - the first part relates differentiation and integration, while the second part provides a method for finding the derivative of a function using the antiderivative of its derivative. This is known as the Fundamental Theorem of Calculus Part II.

4. Can the Fundamental Theorem of Calculus be used for both definite and indefinite integrals?

Yes, the Fundamental Theorem of Calculus applies to both definite and indefinite integrals. For definite integrals, it allows us to evaluate the integral by finding the antiderivative of the function being integrated. For indefinite integrals, it allows us to find the derivative of the function by using its antiderivative.

5. Are there any limitations or exceptions to the Fundamental Theorem of Calculus?

While the Fundamental Theorem of Calculus is a powerful concept, there are some limitations and exceptions. It can only be applied to continuous functions, and it may not work for functions that do not have a well-defined antiderivative. Additionally, the function being integrated must be defined on the entire interval of integration.

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