AP Calculus BC Flashcards: Accumulation Functions Definite Intervals Applied Contexts

What this deck covers

This deck focuses on Accumulation Functions Definite Intervals Applied Contexts, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

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: How does $\int_a^b f(x)\,dx$ change if $a$ and $b$ are swapped?

Answer: It becomes $-\int_b^a f(x)\,dx$. Reversing limits changes sign.

Flashcard 2: Calculate $\int_0^2 (2x + 1)\,dx$.

Answer: $\int_0^2 (2x + 1)\,dx = 6$. Antiderivative: $[x^2 + x]_0^2 = 4 + 2 = 6$.

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

Answer: $\int_0^2 (x^2 + 3)\,dx = \frac{26}{3}$. Using $[\frac{x^3}{3} + 3x]_0^2$.

Flashcard 4: What does the definite integral $\int_a^b r(t)\,dt$ represent in economics?

Answer: Total accumulated revenue over time from $a$ to $b$. Integrating revenue rate gives total revenue.

Flashcard 5: Find $A(x)$ given $f(t) = e^t$ and $a = 0$.

Answer: $A(x) = \int_0^x e^t\,dt = e^x - 1$. Using $[e^t]_0^x = e^x - e^0$.

Flashcard 6: Express $\int_a^b f(x)\,dx$ using the limit of sums.

Answer: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$. Riemann sum definition of definite integral.

Flashcard 7: What does $\int_0^T f(t)\,dt$ represent in population models?

Answer: Total population change over time $0$ to $T$. Integrating population growth rate.

Flashcard 8: What does $\int_0^T f(t)\,dt$ represent in population models?

Answer: Total population change over time $0$ to $T$. Integrating population growth rate.

Flashcard 9: What does the Fundamental Theorem of Calculus Part 2 state?

Answer: If $f$ is continuous on $[a, b]$, $F(x) = \int_a^x f(t)\,dt$ is continuous and differentiable. Shows accumulation functions are differentiable.

Flashcard 10: Evaluate $\int_0^4 (x^3 - 2x)\,dx$.

Answer: $\int_0^4 (x^3 - 2x)\,dx = 48$. Using $[\frac{x^4}{4} - x^2]_0^4 = 64 - 16$.

Flashcard 11: What is the geometric interpretation of $\int_a^b f(x)\,dx$ when $f(x) \geq 0$?

Answer: It is the area under the curve $f(x)$ from $x = a$ to $x = b$. Region bounded by curve and x-axis.

Flashcard 12: Find the accumulation function $A(x)$ given $f(t) = 3t^2$ and $a = 1$.

Answer: $A(x) = \int_1^x 3t^2\,dt = x^3 - 1$. Using FTC: $\int_1^x 3t^2\,dt = [t^3]_1^x$.

Flashcard 13: Find $A(x)$ given $f(t) = e^t$ and $a = 0$.

Answer: $A(x) = \int_0^x e^t\,dt = e^x - 1$. Using $[e^t]_0^x = e^x - e^0$.

Flashcard 14: Determine the value of $\int_1^3 x^2\,dx$.

Answer: $\int_1^3 x^2\,dx = 26/3$. Using $[\frac{x^3}{3}]_1^3 = 9 - \frac{1}{3}$.

Flashcard 15: How do you interpret $\int_a^b v(t)\,dt$ in a physics context?

Answer: It represents the displacement of an object from time $a$ to $b$ given velocity $v(t)$.. Integrating velocity gives position change.

Flashcard 16: Evaluate $\int_0^\pi \sin x\,dx$.

Answer: $\int_0^\pi \sin x\,dx = 2$. $[-\cos x]_0^\pi = -(-1) - (-1)$.

Flashcard 17: Evaluate $\int_0^4 (x^3 - 2x)\,dx$.

Answer: $\int_0^4 (x^3 - 2x)\,dx = 48$. Using $[\frac{x^4}{4} - x^2]_0^4 = 64 - 16$.

Flashcard 18: What is the result of $\int_0^1 5\,dx$?

Answer: 5, since the integral of a constant is the constant times the interval length. Constant $5$ over interval length $1$.

Flashcard 19: Evaluate the integral of a constant, $\int_a^b c\,dx$.

Answer: $c(b-a)$, where $c$ is a constant. Constant times interval length.

Flashcard 20: Evaluate $\int_1^2 \frac{1}{x}\,dx$.

Answer: $\int_1^2 \frac{1}{x}\,dx = \ln 2$. Natural logarithm is antiderivative of $\frac{1}{x}$.

Flashcard 21: State the integration property for $\int_a^b [f(x) + g(x)]\,dx$.

Answer: $\int_a^b f(x)\,dx + \int_a^b g(x)\,dx$. Linearity property of integrals.

Flashcard 22: What is the integral of a function over an interval with zero length?

Answer: Zero, since $\int_a^a f(x)\,dx = 0$. Degenerate interval has no area.

Flashcard 23: Express $\int_a^b f(x)\,dx$ using the limit of sums.

Answer: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$. Riemann sum definition of definite integral.

Flashcard 24: State the relationship between definite integral and area under a curve.

Answer: The definite integral $\int_a^b f(x)\,dx$ represents the net area between $f(x)$ and the x-axis. Positive areas above axis, negative below.

Flashcard 25: Evaluate $\int_0^\pi \sin x\,dx$.

Answer: $\int_0^\pi \sin x\,dx = 2$. $[-\cos x]_0^\pi = -(-1) - (-1)$.

Flashcard 26: Identify the expression for the average value of $f(x)$ on $[a, b]$.

Answer: The average value is $\frac{1}{b-a} \int_a^b f(x)\,dx$. Divides total area by interval width.

Flashcard 27: What does $\frac{d}{dx} \int_a^x f(t)\,dt$ equal?

Answer: $f(x)$, according to the Fundamental Theorem of Calculus Part 2. Derivative of accumulation function equals integrand.

Flashcard 28: What is the formula for accumulation function $A(x)$?

Answer: $A(x) = \int_a^x f(t)\,dt$ where $a$ is a constant and $f(t)$ is a given function. Accumulates $f(t)$ from fixed point $a$ to variable $x$.

Flashcard 29: What is the integral of $\int_a^b kf(x)\,dx$ where $k$ is constant?

Answer: $k \int_a^b f(x)\,dx$. Constant factor property of integrals.

Flashcard 30: Determine $\int_{-1}^1 x^3\,dx$.

Answer: $\int_{-1}^1 x^3\,dx = 0$. Odd function over symmetric interval cancels.

Flashcard 31: What is the Fundamental Theorem of Calculus Part 1?

Answer: If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. Connects antiderivatives to definite integrals.

Flashcard 32: Determine the integral $\int_0^3 4x\,dx$.

Answer: $\int_0^3 4x\,dx = 18$. Using $[2x^2]_0^3 = 2 \cdot 9$.

Flashcard 33: Determine the integral $\int_0^3 4x\,dx$.

Answer: $\int_0^3 4x\,dx = 18$. Using $[2x^2]_0^3 = 2 \cdot 9$.

Flashcard 34: Identify the expression for the average value of $f(x)$ on $[a, b]$.

Answer: The average value is $\frac{1}{b-a} \int_a^b f(x)\,dx$. Divides total area by interval width.

Flashcard 35: What does the Fundamental Theorem of Calculus Part 2 state?

Answer: If $f$ is continuous on $[a, b]$, $F(x) = \int_a^x f(t)\,dt$ is continuous and differentiable. Shows accumulation functions are differentiable.

Flashcard 36: Find the accumulation function $A(x)$ given $f(t) = 3t^2$ and $a = 1$.

Answer: $A(x) = \int_1^x 3t^2\,dt = x^3 - 1$. Using FTC: $\int_1^x 3t^2\,dt = [t^3]_1^x$.

Flashcard 37: Find $\int_0^1 e^x\,dx$.

Answer: $\int_0^1 e^x\,dx = e - 1$. Using $[e^x]_0^1 = e^1 - e^0$.

Flashcard 38: What does the definite integral $\int_a^b r(t)\,dt$ represent in economics?

Answer: Total accumulated revenue over time from $a$ to $b$. Integrating revenue rate gives total revenue.

Flashcard 39: Determine the value of $\int_1^3 x^2\,dx$.

Answer: $\int_1^3 x^2\,dx = 26/3$. Using $[\frac{x^3}{3}]_1^3 = 9 - \frac{1}{3}$.

Flashcard 40: What is the integral of $\int_a^b kf(x)\,dx$ where $k$ is constant?

Answer: $k \int_a^b f(x)\,dx$. Constant factor property of integrals.

Flashcard 41: Evaluate $\int_1^2 \frac{1}{x}\,dx$.

Answer: $\int_1^2 \frac{1}{x}\,dx = \ln 2$. Natural logarithm is antiderivative of $\frac{1}{x}$.

Flashcard 42: What is $\int_a^a f(x)\,dx$ equal to?

Answer: Zero, since the interval has zero length. No area when endpoints are identical.

Flashcard 43: How do you calculate the total distance traveled from $t = a$ to $t = b$?

Answer: $\int_a^b |v(t)|\,dt$ for velocity function $v(t)$.. Absolute value ensures positive distances.

Flashcard 44: Find the value of $\int_0^2 (2 + x)\,dx$.

Answer: $\int_0^2 (2 + x)\,dx = 6$. Using $[2x + \frac{x^2}{2}]_0^2$.

Flashcard 45: How does $\int_a^b f(x)\,dx$ change if $a$ and $b$ are swapped?

Answer: It becomes $-\int_b^a f(x)\,dx$. Reversing limits changes sign.

Flashcard 46: Calculate $\int_0^2 (2x + 1)\,dx$.

Answer: $\int_0^2 (2x + 1)\,dx = 6$. Antiderivative: $[x^2 + x]_0^2 = 4 + 2 = 6$.

Flashcard 47: What is the integral of a function over an interval with zero length?

Answer: Zero, since $\int_a^a f(x)\,dx = 0$. Degenerate interval has no area.

Flashcard 48: What is the formula for accumulation function $A(x)$?

Answer: $A(x) = \int_a^x f(t)\,dt$ where $a$ is a constant and $f(t)$ is a given function. Accumulates $f(t)$ from fixed point $a$ to variable $x$.

Flashcard 49: What is the accumulation function for $f(t) = \cos t$ from $a = 0$?

Answer: $A(x) = \int_0^x \cos t\,dt = \sin x$. Using $[\sin t]_0^x = \sin x - 0$.

Flashcard 50: Express the net change of a quantity using integrals.

Answer: Net change is $\int_a^b F'(x)\,dx = F(b) - F(a)$. FTC relates rate of change to total change.

Flashcard 51: Find $\int_0^2 (x^2 + 3)\,dx$.

Answer: $\int_0^2 (x^2 + 3)\,dx = \frac{26}{3}$. Using $[\frac{x^3}{3} + 3x]_0^2$.

Flashcard 52: What is the result of $\int_0^1 5\,dx$?

Answer: 5, since the integral of a constant is the constant times the interval length. Constant $5$ over interval length $1$.

Flashcard 53: What is $\int_a^a f(x)\,dx$ equal to?

Answer: Zero, since the interval has zero length. No area when endpoints are identical.

Flashcard 54: What is the Fundamental Theorem of Calculus Part 1?

Answer: If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. Connects antiderivatives to definite integrals.

Flashcard 55: What is the geometric interpretation of $\int_a^b f(x)\,dx$ when $f(x) \geq 0$?

Answer: It is the area under the curve $f(x)$ from $x = a$ to $x = b$. Region bounded by curve and x-axis.

Flashcard 56: How do you interpret $\int_a^b v(t)\,dt$ in a physics context?

Answer: It represents the displacement of an object from time $a$ to $b$ given velocity $v(t)$.. Integrating velocity gives position change.

Flashcard 57: How do you calculate the total distance traveled from $t = a$ to $t = b$?

Answer: $\int_a^b |v(t)|\,dt$ for velocity function $v(t)$.. Absolute value ensures positive distances.

Flashcard 58: What is the accumulation function for $f(t) = \cos t$ from $a = 0$?

Answer: $A(x) = \int_0^x \cos t\,dt = \sin x$. Using $[\sin t]_0^x = \sin x - 0$.

Flashcard 59: What does $\frac{d}{dx} \int_a^x f(t)\,dt$ equal?

Answer: $f(x)$, according to the Fundamental Theorem of Calculus Part 2. Derivative of accumulation function equals integrand.

Flashcard 60: State the integration property for $\int_a^b [f(x) + g(x)]\,dx$.

Answer: $\int_a^b f(x)\,dx + \int_a^b g(x)\,dx$. Linearity property of integrals.

Flashcard 61: Find $\int_0^1 e^x\,dx$.

Answer: $\int_0^1 e^x\,dx = e - 1$. Using $[e^x]_0^1 = e^1 - e^0$.

Flashcard 62: Determine $\int_{-1}^1 x^3\,dx$.

Answer: $\int_{-1}^1 x^3\,dx = 0$. Odd function over symmetric interval cancels.

Flashcard 63: Find the value of $\int_0^2 (2 + x)\,dx$.

Answer: $\int_0^2 (2 + x)\,dx = 6$. Using $[2x + \frac{x^2}{2}]_0^2$.

Flashcard 64: Express the net change of a quantity using integrals.

Answer: Net change is $\int_a^b F'(x)\,dx = F(b) - F(a)$.. FTC relates rate of change to total change.

Flashcard 65: Evaluate the integral of a constant, $\int_a^b c\,dx$.

Answer: $c(b-a)$, where $c$ is a constant. Constant times interval length.

Flashcard 66: State the relationship between definite integral and area under a curve.

Answer: The definite integral $\int_a^b f(x)\,dx$ represents the net area between $f(x)$ and the x-axis. Positive areas above axis, negative below.