AP Calculus AB Flashcards: Fundamental Theorem Of Calculus Accumulation Functions

What this deck covers

This deck focuses on Fundamental Theorem Of Calculus Accumulation Functions, 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: Find $\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt$ using the chain rule.

Answer: $f(g(x))g'(x)$. Chain rule applied to variable upper limit of integration.

Flashcard 2: What is the result of $\int_{a}^{b} f(x) \, dx = 0$?

Answer: The areas above and below the x-axis are equal. Net area is zero when positive and negative areas cancel.

Flashcard 3: What does the Fundamental Theorem of Calculus connect?

Answer: Differentiation and integration. The FTC bridges these two fundamental operations.

Flashcard 4: Find $\int_{-2}^{2} x^3 \, dx$.

Answer:

1. $x^3$ is odd, so integral over symmetric interval is zero.

Flashcard 5: What is the Fundamental Theorem of Calculus, Part 1?

Answer: If $F$ is antiderivative of $f$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. This connects antiderivatives with definite integrals.

Flashcard 6: What is the value of $\int_{-a}^{a} f(x) \, dx$ if $f$ is an odd function?

Answer:

1. Odd functions have symmetric areas that cancel over symmetric intervals.

Flashcard 7: Find $\int_{-2}^{2} x^3 \, dx$.

Answer:

1. $x^3$ is odd, so integral over symmetric interval is zero.

Flashcard 8: Evaluate $\int_{1}^{4} \frac{1}{x} \, dx$.

Answer: $\ln 4$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$; evaluate: $\ln(4) - \ln(1) = \ln(4)$.

Flashcard 9: State the antiderivative of $f(x) = \frac{1}{x}$.

Answer: $F(x) = \ln |x| + C$. Standard antiderivative formula for reciprocal function.

Flashcard 10: What does $\int_{a}^{a} f(x) \, dx$ equal for any function $f$?

Answer:

1. When integration limits are equal, the integral is zero.

Flashcard 11: What is the result of $\int_{a}^{b} f(x) \, dx = 0$?

Answer: The areas above and below the x-axis are equal. Net area is zero when positive and negative areas cancel.

Flashcard 12: What property does $\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$ illustrate?

Answer: Reversal of limits changes the sign. Fundamental property showing orientation matters.

Flashcard 13: Find $\frac{d}{dx} \int_{2}^{x} (3t^2 + 2) \, dt$.

Answer: $3x^2 + 2$. By FTC Part 2, derivative of integral equals the integrand.

Flashcard 14: What condition must a function $f$ meet to use the Fundamental Theorem of Calculus?

Answer: $f$ must be continuous on $[a, b]$. Continuity ensures the theorem applies.

Flashcard 15: State the effect of swapping integration limits on the integral value.

Answer: It negates the integral. Swapping limits introduces a negative sign.

Flashcard 16: What condition must a function $f$ meet to use the Fundamental Theorem of Calculus?

Answer: $f$ must be continuous on $[a, b]$. Continuity ensures the theorem applies.

Flashcard 17: Evaluate $\int_{0}^{2} x \, dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Antiderivative is $\frac{x^2}{2}$; evaluate: $\frac{4}{2} - 0 = 2$.

Flashcard 18: Find $\frac{d}{dx} \int_{2}^{x} (3t^2 + 2) \, dt$.

Answer: $3x^2 + 2$. By FTC Part 2, derivative of integral equals the integrand.

Flashcard 19: Evaluate $\int_{1}^{4} \frac{1}{x} \, dx$.

Answer: $\ln 4$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$; evaluate: $\ln(4) - \ln(1) = \ln(4)$.

Flashcard 20: State the relationship between definite integrals and net area.

Answer: Definite integrals represent the net area under a curve. Positive area above x-axis minus negative area below.

Flashcard 21: What is the geometric interpretation of $\int_{a}^{b} f(x) \, dx$?

Answer: Net area between $f(x)$ and the x-axis from $x=a$ to $x=b$. Signed area between curve and x-axis over the interval.

Flashcard 22: What is $\int_{a}^{b} [f(x) + g(x)] \, dx$?

Answer: $\int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$. Linearity property of integration.

Flashcard 23: Evaluate $\int_{1}^{3} (2x+1) \, dx$.

Answer:

1. Antiderivative is $x^2 + x$; evaluate: $(9+3) - (1+1) = 10$.

Flashcard 24: What is the integral of $x^n$ from $a$ to $b$ for $n \neq -1$?

Answer: $\int_{a}^{b} x^n \, dx = \frac{1}{n+1}(b^{n+1} - a^{n+1})$. Apply power rule for integration to definite integrals.

Flashcard 25: What is the geometric interpretation of $\int_{a}^{b} f(x) \, dx$?

Answer: Net area between $f(x)$ and the x-axis from $x=a$ to $x=b$. Signed area between curve and x-axis over the interval.

Flashcard 26: Identify the error: $\int_{a}^{b} f(x) \, dx = F(b) + F(a)$ where $F$ is an antiderivative of $f$.

Answer: Correct: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. Should subtract, not add: $F(b) - F(a)$.

Flashcard 27: Find $\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt$ using the chain rule.

Answer: $f(g(x))g'(x)$. Chain rule applied to variable upper limit of integration.

Flashcard 28: What is the Fundamental Theorem of Calculus, Part 2?

Answer: If $f$ is continuous on $[a, b]$, then $g(x) = \int_{a}^{x} f(t) \, dt$ is differentiable and $g'(x) = f(x)$. Shows that differentiation undoes integration for accumulation functions.

Flashcard 29: Determine $\int_{1}^{2} (2x - 1) \, dx$ using an antiderivative.

Answer:

1. Antiderivative is $x^2 - x$; evaluate: $(4-2) - (1-1) = 1$.

Flashcard 30: What is the meaning of $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$?

Answer: Additivity over intervals. Integrals can be split at any intermediate point.

Flashcard 31: Find $\frac{d}{dx} \int_{0}^{x} e^t \, dt$.

Answer: $e^x$. By FTC Part 2, derivative equals the integrand.

Flashcard 32: What is the value of $\int_{-a}^{a} f(x) \, dx$ if $f$ is an odd function?

Answer:

1. Odd functions have symmetric areas that cancel over symmetric intervals.

Flashcard 33: Evaluate $\int_{0}^{4} (3x^2) \, dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Antiderivative is $x^3$; evaluate: $64 - 0 = 64$.

Flashcard 34: Evaluate $\int_{0}^{\pi} \sin x \, dx$.

Answer:

1. Antiderivative of $\sin x$ is $-\cos x$; evaluate: $-\cos(\pi) - (-\cos(0)) = 2$.

Flashcard 35: State the relationship between definite integrals and net area.

Answer: Definite integrals represent the net area under a curve. Positive area above x-axis minus negative area below.

Flashcard 36: What is the integral of a constant $c$ from $a$ to $b$?

Answer: $\int_{a}^{b} c \, dx = c(b-a)$. Constant times the width of the interval.

Flashcard 37: What property does $\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$ illustrate?

Answer: Reversal of limits changes the sign. Fundamental property showing orientation matters.

Flashcard 38: Find $\frac{d}{dx} \int_{0}^{x} e^t \, dt$.

Answer: $e^x$. By FTC Part 2, derivative equals the integrand.

Flashcard 39: Evaluate $\int_{0}^{2} x \, dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Antiderivative is $\frac{x^2}{2}$; evaluate: $\frac{4}{2} - 0 = 2$.

Flashcard 40: Identify the error: $\int_{a}^{b} f(x) \, dx = F(b) + F(a)$ where $F$ is an antiderivative of $f$.

Answer: Correct: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. Should subtract, not add: $F(b) - F(a)$.

Flashcard 41: What is the integral of $e^x$ from $a$ to $b$?

Answer: $\int_{a}^{b} e^x \, dx = e^b - e^a$. Standard exponential function integration formula.

Flashcard 42: State the effect of swapping integration limits on the integral value.

Answer: It negates the integral. Swapping limits introduces a negative sign.

Flashcard 43: Evaluate $\int_{0}^{3} (4x + 1) \, dx$ using the Fundamental Theorem of Calculus.

Answer: $\frac{45}{2}$. Find antiderivative $2x^2 + x$, evaluate at limits: $(18+3) - (0) = \frac{45}{2}$.

Flashcard 44: Determine $\int_{0}^{1} (2x^3 + 3x^2) \, dx$.

Answer: $\frac{5}{2}$. Antiderivative is $\frac{x^4}{2} + x^3$; evaluate at limits: $\frac{5}{2}$.

Flashcard 45: What is the meaning of $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$?

Answer: Additivity over intervals. Integrals can be split at any intermediate point.

Flashcard 46: What is the Fundamental Theorem of Calculus, Part 1?

Answer: If $F$ is antiderivative of $f$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$. This connects antiderivatives with definite integrals.

Flashcard 47: What does it imply if $\int_{a}^{b} f(x) \, dx > 0$?

Answer: Net area above the x-axis is greater than below. Positive integral means more area above than below x-axis.

Flashcard 48: Evaluate $\int_{0}^{\pi} \sin x \, dx$.

Answer:

1. Antiderivative of $\sin x$ is $-\cos x$; evaluate: $-\cos(\pi) - (-\cos(0)) = 2$.

Flashcard 49: Evaluate $\int_{0}^{3} (4x + 1) \, dx$ using the Fundamental Theorem of Calculus.

Answer: $\frac{45}{2}$. Find antiderivative $2x^2 + x$, evaluate at limits: $(18+3) - (0) = \frac{45}{2}$.

Flashcard 50: Determine $\int_{1}^{2} (2x - 1) \, dx$ using an antiderivative.

Answer:

1. Antiderivative is $x^2 - x$; evaluate: $(4-2) - (1-1) = 1$.

Flashcard 51: What is the integral of a constant $c$ from $a$ to $b$?

Answer: $\int_{a}^{b} c \, dx = c(b-a)$. Constant times the width of the interval.

Flashcard 52: What does the Fundamental Theorem of Calculus connect?

Answer: Differentiation and integration. The FTC bridges these two fundamental operations.

Flashcard 53: State the antiderivative of $f(x) = \frac{1}{x}$.

Answer: $F(x) = \ln |x| + C$. Standard antiderivative formula for reciprocal function.

Flashcard 54: What does $\int_{a}^{a} f(x) \, dx$ equal for any function $f$?

Answer:

1. When integration limits are equal, the integral is zero.

Flashcard 55: What is $\int_{a}^{b} [f(x) + g(x)] \, dx$?

Answer: $\int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx$. Linearity property of integration.

Flashcard 56: State the formula for the derivative of an accumulation function $G(x) = \int_{a}^{x} f(t) \, dt$.

Answer: $G'(x) = f(x)$ if $f$ is continuous. Direct application of FTC Part 2.

Flashcard 57: Evaluate $\int_{1}^{3} (2x+1) \, dx$.

Answer:

1. Antiderivative is $x^2 + x$; evaluate: $(9+3) - (1+1) = 10$.

Flashcard 58: Evaluate $\int_{0}^{4} (3x^2) \, dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Antiderivative is $x^3$; evaluate: $64 - 0 = 64$.

Flashcard 59: What is $\frac{d}{dx} \int_{x}^{1} t^2 \, dt$?

Answer: $-x^2$. Swapping limits introduces negative sign by FTC.

Flashcard 60: What is the integral of $x^n$ from $a$ to $b$ for $n \neq -1$?

Answer: $\int_{a}^{b} x^n \, dx = \frac{1}{n+1}(b^{n+1} - a^{n+1})$. Apply power rule for integration to definite integrals.

Flashcard 61: Determine $\int_{0}^{1} (2x^3 + 3x^2) \, dx$.

Answer: $\frac{5}{2}$. Antiderivative is $\frac{x^4}{2} + x^3$; evaluate at limits: $\frac{5}{2}$.

Flashcard 62: What is the integral of $e^x$ from $a$ to $b$?

Answer: $\int_{a}^{b} e^x \, dx = e^b - e^a$. Standard exponential function integration formula.

Flashcard 63: What is the Fundamental Theorem of Calculus, Part 2?

Answer: If $f$ is continuous on $[a, b]$, then $g(x) = \int_{a}^{x} f(t) \, dt$ is differentiable and $g'(x) = f(x)$. Shows that differentiation undoes integration for accumulation functions.

Flashcard 64: What does it imply if $\int_{a}^{b} f(x) \, dx > 0$?

Answer: Net area above the x-axis is greater than below. Positive integral means more area above than below x-axis.

Flashcard 65: State the formula for the derivative of an accumulation function $G(x) = \int_{a}^{x} f(t) \, dt$.

Answer: $G'(x) = f(x)$ if $f$ is continuous. Direct application of FTC Part 2.