The Fundamental Theorem of Calculus for the TMUA

The Core Principles of the Fundamental Theorem

The Fundamental Theorem of Calculus is the primary link between the two main branches of calculus: differentiation and integration. While you may already be familiar with the mechanics of definite integration, it is crucial to understand the formal relationship it establishes. Integration can be thought of as the reverse of differentiation, identifying what function must have been differentiated to produce the given expression. When we calculate a definite integral, we are using this relationship to find the net change in an antiderivative over a specific interval.

The Evaluation of Definite Integrals

The first form of the theorem used in the TMUA is the standard method for evaluating definite integrals:

$\int_{a}^{b} f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$

This expression formalises the process of finding an antiderivative $F(x)$, and then substituting the upper limit $b$ and the lower limit $a$ to find the difference. Because any constant $c$ added to $F(x)$ would be subtracted away during this process ($(F(b) + c) - (F(a) + c) = F(b) - F(a)$), the constant of integration is omitted in definite integrals.

Properties of Limits and Contiguous Ranges

The theorem allows for several useful manipulations of integral limits that often simplify complex problems. One significant consequence is that swapping the upper and lower limits of an integral introduces a negative sign:

$\int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx$

This is a logical result of the definition, as $F(b) - F(a) = -(F(a) - F(b))$. Furthermore, the theorem permits the splitting of integrals over contiguous ranges:

$\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$

While it is common to assume that $c$ must lie between $a$ and $b$, this is not strictly necessary. The identity holds for any value of $c$, provided that the function $f(x)$ is defined and integrable over the entire interval involved. You must, however, be cautious if $f(x)$ is undefined at $c$ or any other point within the chosen integration range.

Differentiation of an Integral

The second major form of the Fundamental Theorem of Calculus addresses the differentiation of a function that is defined as an integral:

$\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$

This expression demonstrates that if you integrate a function $f(t)$ from a constant lower limit $a$ to a variable upper limit $x$, and then differentiate the resulting expression with respect to $x$, you return to the original function $f(x)$.

Note the change in notation: we use $t$ as a dummy variable inside the integral to avoid confusion with the limit $x$. If the upper limit were a constant instead of a variable, the integral would evaluate to a constant number, and its derivative would simply be zero. This specific form of the theorem is vital for solving problems where a function is defined by its rate of accumulation, a concept occasionally tested in more challenging TMUA paper 2 questions.