AP Calculus AB Flashcards: Exploring Accumulations Of Change

What this deck covers

This deck focuses on Exploring Accumulations Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the integral of $f(x) = \sin(x)$ from $0$ to $\pi$?

Answer: $\int_0^{\pi} \sin(x)\,dx = 2$. Antiderivative is $-\cos(x)$, giving total area under sine curve.

Flashcard 2: What is the integral of $f(x) = 5x$ from $0$ to $3$?

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

Flashcard 3: What is the integral of $f(x) = 4x^3$ from $0$ to $1$?

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

Flashcard 4: What is the integral of $f(x) = 2x$ from $1$ to $3$?

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

Flashcard 5: Find the antiderivative of $f(x) = 5x^4$.

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

Flashcard 6: State the formula for the area under a curve using integration.

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

Flashcard 7: What is the integral of $f(x) = e^x$ from $0$ to $1$?

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

Flashcard 8: What is the integral of $f(x) = \cos(x)$ from $0$ to $\frac{\pi}{2}$?

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

Flashcard 9: What is the integral of $f(x) = 1$ from $0$ to $5$?

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

Flashcard 10: State the definition of an antiderivative.

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

Flashcard 11: Compute $\int_0^2 (4x - x^2)\,dx$.

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

Flashcard 12: Find $\frac{d}{dx} \int_0^x e^{t^2}\,dt$.

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

Flashcard 13: Compute $\int_0^1 (x^2 - x + 1)\,dx$.

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

Flashcard 14: Find $\int_0^2 (x^3 - 3x^2 + 3x)\,dx$.

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

Flashcard 15: Evaluate $\int_0^1 (x^4 - x^2 + 1)\,dx$.

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

Flashcard 16: Compute $\int_0^4 (x - 2)\,dx$.

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

Flashcard 17: Find $\frac{d}{dx} \int_0^x \ln(t)\,dt$.

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

Flashcard 18: Evaluate $\int_1^2 (x^2 + x)\,dx$.

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

Flashcard 19: State the Mean Value Theorem for Integrals.

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

Flashcard 20: Compute $\int_0^2 (3x^2 + 2x)\,dx$.

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

Flashcard 21: State the definition of a definite integral.

Answer: 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.

Flashcard 22: What is the integral of $f(x) = 3x^2$ from $1$ to $4$?

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

Flashcard 23: Find $\frac{d}{dx} \int_0^x \sin(t)\,dt$.

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

Flashcard 24: What is the Fundamental Theorem of Calculus, Part 2?

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

Flashcard 25: What is the integral of $f(x) = x^3$ from $-1$ to $1$?

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

Flashcard 26: What is the average value of $f(x) = x^2$ on $[0, 3]$?

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

Flashcard 27: Evaluate $\int_0^{\pi} \sin(x)\,dx$.

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

Flashcard 28: What is the integral of $f(x) = 3$ from $0$ to $2$?

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

Flashcard 29: What is the integral of $f(x) = e^x$ from $0$ to $1$?

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

Flashcard 30: State the formula for the area under a curve using integration.

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