AP Calculus AB Flashcards: Fundamental Theorem Of Calculus Definite Intervals

What this deck covers

This deck focuses on Fundamental Theorem Of Calculus Definite Intervals, 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 does the second part of the Fundamental Theorem allow us to compute?

Answer: The derivative of an integral function. It gives the rate of change of the area function.

Flashcard 2: Determine $\int_1^4 \frac{1}{x}\,dx$.

Answer: $\ln(4)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$.

Flashcard 3: Find the value of $\int_0^\pi \cos(x)\,dx$.

Answer:

1. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

Flashcard 4: Evaluate $\int_0^4 (3x^2 + 2)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $x^3 + 2x$, evaluate at bounds $4$ and $0$.

Flashcard 5: For $F(x) = \int_0^x \,\sin(t)\,dt$, find $F'(x)$.

Answer: $\sin(x)$. By Part 2 of FTC, derivative equals the integrand.

Flashcard 6: What is the result of $\int_0^2 1\,dx$?

Answer:

1. Integral of constant $1$ over interval length $2$.

Flashcard 7: Determine $\int_1^3 x^2\,dx$ using an antiderivative.

Answer: $26/3$. Use antiderivative $\frac{x^3}{3}$, then evaluate at bounds.

Flashcard 8: Evaluate $\int_0^3 (5x - 4)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $\frac{5x^2}{2} - 4x$, evaluate at bounds.

Flashcard 9: What is a definite integral's geometric interpretation?

Answer: Net area between the curve and the x-axis over $[a, b]$. Represents the signed area between curve and x-axis.

Flashcard 10: For $F(x) = \int_0^x e^t\,dt$, find $F'(x)$.

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

Flashcard 11: What is the integral of $f(x) = 1/x$ over $[1, e]$?

Answer:

1. Antiderivative of $\frac{1}{x}$ is $\ln(x)$; $\ln(e) - \ln(1) = 1$.

Flashcard 12: What condition must a function $f$ satisfy for the Fundamental Theorem to apply?

Answer: $f$ must be continuous on $[a, b]$. Continuity ensures the antiderivative exists and is differentiable.

Flashcard 13: State the Fundamental Theorem of Calculus, Part 2.

Answer: If $f$ is continuous on $[a,b]$, then $F(x) = \int_a^x f(t)\,dt$ is differentiable and $F'(x) = f(x)$. Shows that differentiation and integration are inverse operations.

Flashcard 14: Determine $\int_0^2 x^2\,dx$ using an antiderivative.

Answer: $\frac{8}{3}$. Use antiderivative $\frac{x^3}{3}$, evaluate at bounds.

Flashcard 15: Find $\int_0^4 (x^2 - 2)\,dx$ using the Fundamental Theorem.

Answer: $\frac{32}{3}$. Use antiderivative $\frac{x^3}{3} - 2x$, evaluate at bounds.

Flashcard 16: Determine $\int_1^4 \frac{1}{x}\,dx$.

Answer: $\ln(4)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$.

Flashcard 17: Define a proper antiderivative.

Answer: A function $F(x)$ whose derivative is $f(x)$. An antiderivative is any function whose derivative gives $f(x)$.

Flashcard 18: What is the result of a definite integral when the upper and lower limits are equal?

Answer:

1. When limits are equal, the interval has zero length.

Flashcard 19: Evaluate $\int_0^3 (5x - 4)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $\frac{5x^2}{2} - 4x$, evaluate at bounds.

Flashcard 20: Determine $\int_0^2 x^2\,dx$ using an antiderivative.

Answer: $\frac{8}{3}$. Use antiderivative $\frac{x^3}{3}$, evaluate at bounds.

Flashcard 21: State the Fundamental Theorem of Calculus, Part 1.

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

Flashcard 22: Find $\int_0^\pi \sin(x)\,dx$ using the Fundamental Theorem.

Answer:

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

Flashcard 23: What does the Fundamental Theorem of Calculus connect?

Answer: It connects differentiation and integration. They are inverse operations of each other.

Flashcard 24: Find $\int_0^2 (4x^3 + x)\,dx$ using an antiderivative.

Answer:

1. Use antiderivative $x^4 + \frac{x^2}{2}$, evaluate at bounds.

Flashcard 25: Evaluate $\int_{-2}^2 (x^3 + 2x)\,dx$ using symmetry properties.

Answer: $0$. Both functions are odd, so integral over symmetric interval is zero.

Flashcard 26: Find the derivative of $F(x) = \int_2^x \ln(t)\,dt$.

Answer: $\ln(x)$. By Part 2 of FTC, the derivative equals the integrand.

Flashcard 27: Calculate $\int_1^3 (2x^2 - x)\,dx$ using the Fundamental Theorem.

Answer: $\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.

Flashcard 28: Calculate $\int_0^1 (x^3 + 1)\,dx$ using the Fundamental Theorem.

Answer: $\frac{5}{4}$. Use antiderivative $\frac{x^4}{4} + x$, evaluate at bounds.

Flashcard 29: Evaluate $\int_0^4 (3x^2 + 2)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $x^3 + 2x$, evaluate at bounds $4$ and $0$.

Flashcard 30: Compute $\int_{-1}^1 x^3\,dx$ using symmetry properties.

Answer:

1. Odd function over symmetric interval gives zero area.

Flashcard 31: What is an antiderivative of $f(x) = 3x^2$?

Answer: $F(x) = x^3 + C$. Power rule: increase exponent by 1, divide by new exponent.

Flashcard 32: Find the value of $\int_0^\pi \cos(x)\,dx$.

Answer:

1. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

Flashcard 33: Calculate $\int_0^1 (x^3 + 1)\,dx$ using the Fundamental Theorem.

Answer: $\frac{5}{4}$. Use antiderivative $\frac{x^4}{4} + x$, evaluate at bounds.

Flashcard 34: What is the definite integral of a constant $c$ over an interval $[a, b]$?

Answer: $c(b-a)$. A constant function integrates to constant times interval length.

Flashcard 35: Find $\int_0^\pi \sin(x)\,dx$ using the Fundamental Theorem.

Answer:

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

Flashcard 36: Evaluate $\int_0^3 (x^2 + x + 1)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $\frac{x^3}{3} + \frac{x^2}{2} + x$, evaluate at bounds.

Flashcard 37: Evaluate $\int_0^1 (4x^3 - x)\,dx$ using the Fundamental Theorem.

Answer: $\frac{3}{4}$. Use antiderivative $x^4 - \frac{x^2}{2}$, evaluate at bounds.

Flashcard 38: Find $\int_0^4 (x^2 - 2)\,dx$ using the Fundamental Theorem.

Answer: $\frac{32}{3}$. Use antiderivative $\frac{x^3}{3} - 2x$, evaluate at bounds.

Flashcard 39: Evaluate $\int_0^2 (3x^2 - 2x)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $x^3 - x^2$, evaluate at bounds.

Flashcard 40: What is the result of $\int_0^2 1\,dx$?

Answer:

1. Integral of constant $1$ over interval length $2$.

Flashcard 41: What condition must a function $f$ satisfy for the Fundamental Theorem to apply?

Answer: $f$ must be continuous on $[a, b]$. Continuity ensures the antiderivative exists and is differentiable.

Flashcard 42: What is the purpose of the definite integral?

Answer: To find the net area under a curve. It represents the signed area between curve and x-axis.

Flashcard 43: Evaluate $\int_0^2 (3x^2 - 2x)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $x^3 - x^2$, evaluate at bounds.

Flashcard 44: What is the purpose of the definite integral?

Answer: To find the net area under a curve. It represents the signed area between curve and x-axis.

Flashcard 45: Find the derivative of $F(x) = \int_2^x \ln(t)\,dt$.

Answer: $\ln(x)$. By Part 2 of FTC, the derivative equals the integrand.

Flashcard 46: Compute $\int_{-1}^1 x^3\,dx$ using symmetry properties.

Answer:

1. Odd function over symmetric interval gives zero area.

Flashcard 47: What is the integral of $f(x) = e^x$ over $[0, 1]$?

Answer: $e - 1$. Antiderivative of $e^x$ is $e^x$; evaluate at bounds.

Flashcard 48: What is the result of a definite integral when the upper and lower limits are equal?

Answer:

1. When limits are equal, the interval has zero length.

Flashcard 49: Evaluate $\int_{-2}^2 (x^3 + 2x)\,dx$ using symmetry properties.

Answer: $0$. Both functions are odd, so integral over symmetric interval is zero.

Flashcard 50: Evaluate $\int_0^3 (x^2 + x + 1)\,dx$ using the Fundamental Theorem.

Answer:

1. Use antiderivative $\frac{x^3}{3} + \frac{x^2}{2} + x$, evaluate at bounds.

Flashcard 51: State the Fundamental Theorem of Calculus, Part 2.

Answer: If $f$ is continuous on $[a,b]$, then $F(x) = \int_a^x f(t)\,dt$ is differentiable and $F'(x) = f(x)$. Shows that differentiation and integration are inverse operations.

Flashcard 52: What is an antiderivative of $f(x) = 3x^2$?

Answer: $F(x) = x^3 + C$. Power rule: increase exponent by 1, divide by new exponent.

Flashcard 53: Calculate $\int_1^3 (2x^2 - x)\,dx$ using the Fundamental Theorem.

Answer: $\frac{26}{3}$. Use antiderivative $\frac{2x^3}{3} - \frac{x^2}{2}$, evaluate at bounds.

Flashcard 54: What is the integral of $f(x) = e^x$ over $[0, 1]$?

Answer: $e - 1$. Antiderivative of $e^x$ is $e^x$; evaluate at bounds.

Flashcard 55: What does the second part of the Fundamental Theorem allow us to compute?

Answer: The derivative of an integral function. It gives the rate of change of the area function.

Flashcard 56: What is a definite integral's geometric interpretation?

Answer: Net area between the curve and the x-axis over $[a, b]$. Represents the signed area between curve and x-axis.

Flashcard 57: What does it mean for a function $F(x)$ to be differentiable?

Answer: $F(x)$ has a derivative at every point in its domain. The function has a well-defined derivative at each point.

Flashcard 58: What is the integral of $f(x) = \sin(x)$ over $[0, \pi]$?

Answer:

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

Flashcard 59: Find $\int_0^1 (x^2 + 2x)\,dx$ using an antiderivative.

Answer: $\frac{7}{3}$. Use antiderivative $\frac{x^3}{3} + x^2$, evaluate at bounds.

Flashcard 60: What does the Fundamental Theorem of Calculus connect?

Answer: It connects differentiation and integration. They are inverse operations of each other.

Flashcard 61: What is the integral of $f(x) = \cos(x)$ over $[0, \pi]$?

Answer:

1. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

Flashcard 62: Find $\int_0^1 (x^2 + 2x)\,dx$ using an antiderivative.

Answer: $\frac{7}{3}$. Use antiderivative $\frac{x^3}{3} + x^2$, evaluate at bounds.

Flashcard 63: Evaluate $\int_0^1 2x\,dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Find antiderivative $x^2$, then evaluate $1^2 - 0^2 = 1$.

Flashcard 64: Find $\int_0^2 (4x^3 + x)\,dx$ using an antiderivative.

Answer:

1. Use antiderivative $x^4 + \frac{x^2}{2}$, evaluate at bounds.

Flashcard 65: Determine $\int_1^3 x^2\,dx$ using an antiderivative.

Answer: $26/3$. Use antiderivative $\frac{x^3}{3}$, then evaluate at bounds.

Flashcard 66: Determine $\int_0^1 (5x^2 - 3x)\,dx$ using an antiderivative.

Answer: $\frac{1}{3}$. Use antiderivative $\frac{5x^3}{3} - \frac{3x^2}{2}$, evaluate at bounds.

Flashcard 67: Evaluate $\int_0^1 (4x^3 - x)\,dx$ using the Fundamental Theorem.

Answer: $\frac{3}{4}$. Use antiderivative $x^4 - \frac{x^2}{2}$, evaluate at bounds.

Flashcard 68: What is the definite integral of a constant $c$ over an interval $[a, b]$?

Answer: $c(b-a)$. A constant function integrates to constant times interval length.

Flashcard 69: For $F(x) = \int_0^x e^t\,dt$, find $F'(x)$.

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

Flashcard 70: What is the integral of $f(x) = 1/x$ over $[1, e]$?

Answer:

1. Antiderivative of $\frac{1}{x}$ is $\ln(x)$; $\ln(e) - \ln(1) = 1$.

Flashcard 71: For $F(x) = \int_0^x \,\sin(t)\,dt$, find $F'(x)$.

Answer: $\sin(x)$. By Part 2 of FTC, derivative equals the integrand.

Flashcard 72: What is the integral of $f(x) = \sin(x)$ over $[0, \pi]$?

Answer:

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

Flashcard 73: Determine $\int_0^1 (5x^2 - 3x)\,dx$ using an antiderivative.

Answer: $\frac{1}{3}$. Use antiderivative $\frac{5x^3}{3} - \frac{3x^2}{2}$, evaluate at bounds.

Flashcard 74: Evaluate $\int_0^1 2x\,dx$ using the Fundamental Theorem of Calculus.

Answer:

1. Find antiderivative $x^2$, then evaluate $1^2 - 0^2 = 1$.

Flashcard 75: Define a proper antiderivative.

Answer: A function $F(x)$ whose derivative is $f(x)$. An antiderivative is any function whose derivative gives $f(x)$.

Flashcard 76: What does it mean for a function $F(x)$ to be differentiable?

Answer: $F(x)$ has a derivative at every point in its domain. The function has a well-defined derivative at each point.

Flashcard 77: What is the integral of $f(x) = \cos(x)$ over $[0, \pi]$?

Answer:

1. Antiderivative is $\sin(x)$; $\sin(\pi) - \sin(0) = 0$.

Flashcard 78: State the Fundamental Theorem of Calculus, Part 1.

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