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 $f(x) = \cos(x)$ 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 2: Compute $\int_0^4 (x - 2)\,dx$.

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

Flashcard 3: 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 4: 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 5: 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 6: 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 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: 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 9: Evaluate $\int_1^2 (x^2 + x)\,dx$.

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

Flashcard 10: 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 11: Find $\int_0^2 (x^3 - 3x^2 + 3x)\,dx$.

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

Flashcard 12: 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 13: Compute $\int_0^4 (x - 2)\,dx$.

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

Flashcard 14: 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 15: 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 16: Find $\frac{d}{dx} \int_0^x \ln(t)\,dt$.

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

Flashcard 17: 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 18: Evaluate $\int_0^{\pi} \sin(x)\,dx$.

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

Flashcard 19: 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 20: 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 21: Evaluate $\int_0^1 (x^4 - x^2 + 1)\,dx$.

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

Flashcard 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: Compute $\int_0^2 (4x - x^2)\,dx$.

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

Flashcard 29: 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 30: Compute $\int_0^1 (x^2 - x + 1)\,dx$.

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

Flashcard 31: Evaluate $\int_0^{\pi} \sin(x)\,dx$.

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

Flashcard 32: 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 33: 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 34: 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 35: 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 36: 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 37: 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 38: Evaluate $\int_1^2 (x^2 + x)\,dx$.

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

Flashcard 39: 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 40: 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 41: Compute $\int_0^1 (x^2 - x + 1)\,dx$.

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

Flashcard 42: Find $\int_0^2 (x^3 - 3x^2 + 3x)\,dx$.

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

Flashcard 43: 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 44: 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 45: 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 46: 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 47: 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 48: 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 49: 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 50: 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 51: Compute $\int_0^2 (4x - x^2)\,dx$.

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

Flashcard 52: 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 53: 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 54: Evaluate $\int_0^1 (x^4 - x^2 + 1)\,dx$.

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

Flashcard 55: 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 56: Find $\frac{d}{dx} \int_0^x \ln(t)\,dt$.

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