Fundamental Theorem of Calculus in Engineering Mathematics

The Fundamental Theorem of Calculus (FTC) links differentiation and integration as inverse processes, enabling the evaluation of definite integrals through antiderivatives. It consists of two parts: Part 1 shows that if $f$ is continuous on $[a,b]$, then the function $\displaystyle F(x) = \int_a^x f(t)\,dt$ is differentiable on with derivative . Part 2 states that if is an antiderivative of a continuous function on , then . The theorem requires continuity of to ensure the existence of the antiderivative and integrability over the interval. FTC simplifies computation of areas under curves and accumulated quantities from rates of change, important in engineering fields such as dynamics and control theory. It forms the basis for numerical integration techniques and symbolic integration algorithms. Understanding FTC aids in interpreting physical laws expressed via differential and integral equations and is key for design, analysis, and optimization tasks in engineering.