What would you like to understand?

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., $F' = f$).

The accumulation function $A(x)$ defines area by integrating from a fixed lower endpoint to a variable upper endpoint $x$. For $x \ge 0$, this represents ordinary geometric area.

Conditions: The lower endpoint is fixed.; The upper endpoint $x$ is variable.; Signed integration is used to handle direction and negative values.

The accumulation function $A(x)$ defines area by integrating from a fixed lower endpoint to a variable upper endpoint $x$. For $x \ge 0$, this represents ordinary geometric area.

Conditions: The lower endpoint is fixed.; The upper endpoint $x$ is variable.; Signed integration is used to handle direction and negative values.

The accumulation idea applies to motion by integrating velocity over time. Integrating signed velocity gives net displacement, while integrating the absolute value of velocity gives total distance traveled.

Conditions: The variable of integration is time.; The integrand is velocity (signed or absolute).

The accumulation idea applies to motion by integrating velocity over time. Integrating signed velocity gives net displacement, while integrating the absolute value of velocity gives total distance traveled.

Conditions: The variable of integration is time.; The integrand is velocity (signed or absolute).

To evaluate $\int_2^3 \sqrt{x} dx$, first find the antiderivative $\frac{2}{3}x^{3/2}$. Then, apply the Fundamental Theorem of Calculus by subtracting the value of the antiderivative at the lower bound from its value at the upper bound: $\left[\frac{2}{3}x^{3/2}\right]_2^3 = \frac{2}{3}(3)^{3/2} - \frac{2}{3}(2)^{3/2}$.

Conditions: The integral is $\int_2^3 \sqrt{x} dx$.; The antiderivative is $\frac{2}{3}x^{3/2}$.; The bounds are 2 and 3.

To evaluate $\int_2^3 \sqrt{x} dx$, first find the antiderivative $\frac{2}{3}x^{3/2}$. Then, apply the Fundamental Theorem of Calculus by subtracting the value of the antiderivative at the lower bound from its value at the upper bound: $\left[\frac{2}{3}x^{3/2}\right]_2^3 = \frac{2}{3}(3)^{3/2} - \frac{2}{3}(2)^{3/2}$.

Conditions: The integral is $\int_2^3 \sqrt{x} dx$.; The antiderivative is $\frac{2}{3}x^{3/2}$.; The bounds are 2 and 3.

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