Exploring Accumulations of Change - AP Calculus AB

$\int_0^3 5x,dx = 22.5$. Linear function $5x$ has antiderivative $\frac{5x^2}{2}$.

$\int_0^1 4x^3,dx = 1$. Power rule: antiderivative of $x^3$ is $\frac{x^4}{4}$.

$\int_1^3 2x,dx = 8$. Antiderivative is $x^2$, evaluated from 1 to 3 gives $9-1=8$.

$F(x) = x^5 + C$. Power rule for antiderivatives: increase exponent by 1, divide by new exponent.

Area = $\int_a^b f(x),dx$. Fundamental connection between integration and area under curves.

$\int_0^1 e^x,dx = e - 1$. Antiderivative is $e^x$, evaluated from 0 to 1 gives $e-1$.

$\int_0^{\frac{\pi}{2}} \cos(x),dx = 1$. Antiderivative is $\sin(x)$, evaluated from 0 to $\frac{\pi}{2}$ gives $1-0=1$.

$\int_0^5 1,dx = 5$. Integral of constant function equals constant times interval length.

A function $F(x)$ such that $F'(x) = f(x)$ for all $x$ in the domain. Function whose derivative equals the given function.

$\frac{16}{3}$. Quadratic function forming parabolic region with positive area.

$e^{x^2}$. By FTC Part 1, derivative equals the integrand with $x$ substituted.

$\frac{5}{6}$. Quadratic function integrated using power rule for each term.

$2$. Cubic polynomial integrated using power rule for each term.

$\frac{5}{6}$. Sum of power functions integrated using standard power rule.

$0$. Linear function with zero net area due to symmetry about $x=2$.

$\ln(x)$. By FTC Part 1, derivative of integral equals integrand.

$\frac{7}{3}$. Sum of power functions integrated using power rule.

$\exists c \in [a, b]$ such that $f(c) = \frac{1}{b-a}\int_a^b f(x),dx$. Guarantees existence of point where function equals its average value.

$12$. Antiderivative is $x^3 + x^2$, evaluated from 0 to 2 gives $8 + 4 = 12$.

The limit of Riemann sums: $\lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\Delta x$. Formal definition using limit of approximating rectangular areas.

$\int_1^4 3x^2,dx = 63$. Antiderivative is $x^3$, so $F(4) - F(1) = 64 - 1 = 63$.

$\sin(x)$. By FTC Part 1, derivative of integral with variable upper limit equals integrand.

$\int_a^b f(x),dx = F(b) - F(a)$, where $F$ is an antiderivative of $f$. States that definite integral equals antiderivative evaluated at bounds.

$0$. Odd function over symmetric interval gives zero due to cancellation.

$\frac{1}{3} \int_0^3 x^2,dx = 3$. Average value formula: $\frac{1}{b-a}\int_a^b f(x)dx$ applied to $x^2$ on $[0,3]$.

$2$. Antiderivative is $-\cos(x)$, evaluated from 0 to $\pi$ gives $-(-1)-(-1)=2$.

$\int_0^2 3,dx = 6$. Integral of constant equals constant times interval width.

$\int_0^1 e^x,dx = e - 1$. Antiderivative is $e^x$, evaluated from $0$ to $1$ gives $e-1$.

Area = $\int_a^b f(x),dx$. Fundamental connection between integration and area under curves.