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

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

Flashcard 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: Determine the integral $\int_0^3 4x\,dx$.

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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.