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The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

Continuity ensures that the local average height of the function over a small interval approaches the exact height at the endpoint as the interval width shrinks to zero. This property is crucial for proving that the derivative of the accumulation function $A(x)$ is exactly the integrand $f(x)$, as it allows the replacement of the average value with the point value in the limit.

Conditions: The function $f$ is continuous at the point $x$.; The interval width $h$ approaches zero.

Continuity ensures that the local average height of the function over a small interval approaches the exact height at the endpoint as the interval width shrinks to zero. This property is crucial for proving that the derivative of the accumulation function $A(x)$ is exactly the integrand $f(x)$, as it allows the replacement of the average value with the point value in the limit.

Conditions: The function $f$ is continuous at the point $x$.; The interval width $h$ approaches zero.