1. Fundamental Theorem of Calculus: By the Fundamental Theorem of Calculus, if $F(x)$ is the antiderivative of $f(x)$, then:
$$\int_{a}^{b} f(x)dx = F(b) - F(a)$$
2. Applying the Condition $a=b$: If the lower limit ($a$) is equal to the upper limit ($b$), we substitute $b$ for $a$ in the evaluation formula:
$$\int_{a}^{a} f(x)dx = F(a) - F(a)\lt strong\gt 3. Result:\lt /strong\gt F(a) - F(a) = 0$$
Geometrically, this represents the "area" of a region with zero width. Since area is calculated as $\text{height} \times \text{width}$, and the width is $(a - a) = 0$, the resulting area must be zero regardless of the function's height.