Second Fundamental Theorem of Calculus
The second Fundamental Theorem of Calculus gives a relationship between the two processes of differentiation and integration, which are inverse processes of each other. For a function f(x) the differentiation of the anti-derivative of the function results back in the original function.
Let us learn more about the statement, proof of the second fundamental theorem of calculus, with the help of examples, FAQs.
What Is Second Fundamental Theorem of Calculus?
The Second Fundamental Theorem of Calculus relates the two operations of differentiation and integration and shows that these two operations are inverse of each other. This theorem connects the differentiation of a function which is the gradient of a curve with the integration of the function which is the area enclosed under the curve. The second fundamental theorem states that the differentiation of the anti-derivative of a function gives back the original function.
Statement: If a function f is continuous over the interval [a, b] and differentiable across the interval (a, b) then the differentiation of the anti-derivative of the function gives back the function f.
\(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\)
Proof of Second Fundamental Theorem of Calculus
The proof includes three simple steps. First, the given function f(t) is integrated, secondly, the upper and lower bounds of integration are applied, and finally, the differentiation of this expression gives back the initial function.
\(\begin{align}\dfrac{d}{dx}\left[\int ^x_af(t).dt\right]&=\dfrac{d}{dx}[F(t)]^x_a \\&=\dfrac{d}{dx}[F(x) - F(a)] \\&= F'(x) = f(x)\end{align}\)
The integration of f(t) is equal to F(t). Further, the upper bound limit of x and the lower bound limit of a is applied for the function F(x), to obtain F(x) - F(a). The derivation of F(x) is equal to F'(x), which is equal to f(x), the original function.
Related Topics
The following topics are helpful for a better understanding of the second fundamental theorem of calculus.
FAQs on Second Fundamental Theorem of Calculus
What Is The Second Fundamental Theorem of Calculus?
The second fundamental theorem of calculus gives a holistic relationship between the two processes of integration and differentiation. It states that, if a function f is continuous over the interval [a, b] and differentiable across the interval (a, b) then the differentiation of the anti-derivative of the function gives back the function f. This is expressed in the form of a mathematical expression as \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\).
What Is The Formula Of Second Fundamental Theorem of Calculus?
The formula for the second fundamental theorem of calculus is \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\). Here the differentiation of the anti-derivative of the function f(x) is computed, which is equal to the initial function f(x).
What Is The Use Of Second Fundamental Theorem of Calculus?
The second fundamental theorem of calculus is useful to understand the relationship between the two important operations of differentiation and integration. Integration is also referred as anti-derivative, and this theorem is useful to understand that integration and differentiation are the reverse processes of each other.
What Is The Difference Between First Fundamental Theorem Of Calculus And Second Fundamental Theorem Of Calculus?
The first fundamental theorem of calculus explains the anti-derivative of a function with an upper bound, lower bound, and the second fundamental theorem of calculus gives a relationship between the process of differentiation and integration. The first fundamental theorem of calculus states that the integration or the anti-derivative of a function can be taken with an upper bound and a lower bound. \( [\int^a_b f(x).dx] = F(b) - F(a) \). The second fundamental theorem of calculus states that the differentiation of the anti-derivative of a function gives back the initial function. \(\dfrac{d}{dx}\int^x_af(x).dx = f(x)\)