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This occurs when the integrand is continuous on the relevant interval. Under this condition, the Fundamental Theorem of Calculus states that the derivative of the integral with respect to its upper limit is the integrand evaluated at that limit.

Conditions: The integrand is continuous on the interval of integration.; The upper endpoint is the variable with respect to which we are differentiating (before applying the chain rule for a composite limit).

This occurs when the integrand is continuous on the relevant interval. Under this condition, the Fundamental Theorem of Calculus states that the derivative of the integral with respect to its upper limit is the integrand evaluated at that limit.

Conditions: The integrand is continuous on the interval of integration.; The upper endpoint is the variable with respect to which we are differentiating (before applying the chain rule for a composite limit).

The argument places strips of height $2\pi r$ along the radius axis, forming a Riemann sum. As the maximum partition width tends to zero, these sums approach the area beneath the linear circumference function.

Conditions: The partition is refined such that the maximum width tends to zero.; The circumference function $2\pi r$ is treated as the height of the strips.

The argument places strips of height $2\pi r$ along the radius axis, forming a Riemann sum. As the maximum partition width tends to zero, these sums approach the area beneath the linear circumference function.

Conditions: The partition is refined such that the maximum width tends to zero.; The circumference function $2\pi r$ is treated as the height of the strips.

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.

To verify an antiderivative, you differentiate the proposed result. If the derivative of the proposed function equals the original integrand, then the proposal is correct.

Conditions: The proposed function is differentiable.; The original integrand is defined on the domain of interest.

The indefinite integral of $x^2$ yields a family of functions $F(x) = \frac{x^3}{3} + C$. By defining the accumulation function $A(x)$ such that $A(0) = \int_0^0 t^2 dt = 0$, we impose an initial condition.

Conditions: Integrand is $f(t) = t^2$; Lower limit of integration is 0; Accumulation starts from zero

The indefinite integral of $x^2$ yields a family of functions $F(x) = \frac{x^3}{3} + C$. By defining the accumulation function $A(x)$ such that $A(0) = \int_0^0 t^2 dt = 0$, we impose an initial condition.

Conditions: Integrand is $f(t) = t^2$; Lower limit of integration is 0; Accumulation starts from zero