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

Answer:

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

Flashcard 3: 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 4: 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 5: 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 6: 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 7: Evaluate $\int_{0}^{\pi} \sin x \, dx$.

Answer:

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

Flashcard 8: What is $\frac{d}{dx} \int_{x}^{1} t^2 \, dt$?

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

Flashcard 9: 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 10: Evaluate $\int_{0}^{4} (3x^2) \, dx$ using the Fundamental Theorem of Calculus.

Answer:

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

Flashcard 11: State the effect of swapping integration limits on the integral value.

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

Flashcard 12: 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 13: 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 14: 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 15: 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 16: 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 17: Determine $\int_{1}^{2} (2x - 1) \, dx$ using an antiderivative.

Answer:

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

Flashcard 18: 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 19: Find $\frac{d}{dx} \int_{0}^{x} e^t \, dt$.

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

Flashcard 20: 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 21: 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 22: 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 23: 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 24: What does the Fundamental Theorem of Calculus connect?

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

Flashcard 25: 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 26: 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 27: 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 28: 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 29: 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 30: State the antiderivative of $f(x) = \frac{1}{x}$.

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