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The geometric meaning is that each small horizontal step uses its own local slope. The resulting vertical increments accumulate to the total rise from $f(a)$ to $f(b)$.

Conditions: Viewing the theorem through the lens of Riemann sums or local linear approximation.; The function is differentiable on the interval.

The geometric meaning is that each small horizontal step uses its own local slope. The resulting vertical increments accumulate to the total rise from $f(a)$ to $f(b)$.

Conditions: Viewing the theorem through the lens of Riemann sums or local linear approximation.; The function is differentiable on the interval.

The net-change formula states that for a continuously differentiable function on an interval, integrating its derivative gives the endpoint difference. Specifically, the integral of the derivative $f'$ from $a$ to $b$ equals $f(b) - f(a)$.

Conditions: The function is continuously differentiable on the interval.; The integration is over the interval from $a$ to $b$.

The net-change formula states that for a continuously differentiable function on an interval, integrating its derivative gives the endpoint difference. Specifically, the integral of the derivative $f'$ from $a$ to $b$ equals $f(b) - f(a)$.

Conditions: The function is continuously differentiable on the interval.; The integration is over the interval from $a$ to $b$.

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.

The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.

The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.