steelers2147 said:Homework Statement
Complete the proof by using the Fundamental Theorem of Calculus TWICE to establish
[tex]
\int_c^d(\int_a^b f _{x}(x,y)dx)dy=...=\int_a^b(\int_{c}^{d}f_{x}(x,y)dy)dx
[/tex]
Homework Equations
I know that the FTC states that if g(x)=\int_a^x(f), then g'=f
The Attempt at a Solution
I'm not sure how to use this fact to get the proof started. Any guidance would be appreciated.
The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes the relationship between derivatives and integrals. It states that if a function is continuous on a closed interval [a, b] and f(x) is its antiderivative, then the definite integral of f(x) from a to b is equal to f(b) - f(a).
The Fundamental Theorem of Calculus is used to evaluate definite integrals and to find the area under a curve. It also allows us to find the derivative of a function by using the inverse relationship between derivatives and integrals.
The First Fundamental Theorem of Calculus states the relationship between derivatives and integrals, while the Second Fundamental Theorem of Calculus states the relationship between definite and indefinite integrals. In other words, the first theorem deals with the antiderivative of a function, while the second theorem deals with the area under a curve.
Proving the Fundamental Theorem of Calculus twice is important because it provides a deeper understanding of the concept and its applications. The first proof shows the connection between derivatives and integrals, while the second proof shows how to use this connection to evaluate definite integrals.
The first proof involves using the Mean Value Theorem to show that the definite integral is equal to the antiderivative evaluated at the endpoints. The second proof uses the definition of the definite integral and the Fundamental Theorem of Calculus (part 1) to show that the definite integral is equal to the difference of the antiderivative evaluated at the endpoints. Both proofs require a good understanding of the properties of integrals and derivatives.