Answer
The Fundamental Theorem of Calculus is crucial in calculus because it provides a deep connection between the concepts of differentiation and integration. It allows us to compute definite integrals by finding antiderivatives and simplifies the process of evaluating integrals.
Work Step by Step
The Fundamental Theorem of Calculus (FTC) is a pair of theorems that link the concept of differentiation and integration. There are two parts to the theorem: the first part and the second part.
First Part of the Fundamental Theorem of Calculus:
If \(f\) is a continuous real-valued function on the closed interval \([a, b]\), and \(F\) is the function defined by
\[ F(x) = \int_{a}^{x} f(t) \,dt \]
for all \(x\) in \([a, b]\), then \(F\) is continuous on \([a, b]\) and differentiable on the open interval \((a, b)\), and \(F'(x) = f(x)\) for all \(x\) in \((a, b)\).
1st part example:
Let \(f(x) = 2x\) and consider \(F(x) = \int_{0}^{x} 2t \,dt\). Apply the first part of the FTC:
\[ F(x) = \int_{0}^{x} 2t \,dt = \left[t^2\right]_{0}^{x} = x^2 \]
Now, differentiate \(F(x)\): \(F'(x) = 2x\), which is the original function \(f(x)\).
Second Part of the Fundamental Theorem of Calculus:
If \(f\) is a real-valued function on the closed interval \([a, b]\) and \(F\) is any antiderivative of \(f\) on \([a, b]\), then
\[ \int_{a}^{b} f(x) \,dx = F(b) - F(a) \]
Example:
Let \(f(x) = 2x\) and \(F(x) = x^2\) as before. According to the second part of the FTC:
\[ \int_{0}^{2} 2x \,dx = F(2) - F(0) = 2^2 - 0^2 = 4 \]
This illustrates that the definite integral of \(f(x) = 2x\) from 0 to 2 is equal to the difference of the antiderivative \(F(x) = x^2\) evaluated at the upper and lower limits.
