AP Calculus BC Flashcards: Fundamental Theorem Of Calculus Definite Intervals

Study Fundamental Theorem Of Calculus Definite Intervals in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus BC

Fundamental Theorem Of Calculus Definite Intervals

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QUESTION

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Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

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ANSWER

$\frac{13}{4}$ . Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.

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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 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.

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Flashcard 1: Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

Answer: $\frac{13}{4}$ . Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.

Flashcard 2: Find the integral of $f(x) = 1 + x$ from 0 to 2.

Answer:

Antiderivative is $x + \frac{x^2}{2}$ , evaluate at bounds.

Flashcard 3: Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.

Answer: $\text{ln}(4)$ . Same as $\int \frac{1}{x} dx$ , antiderivative is $\ln(x)$ .

Flashcard 4: Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.

Answer: $\frac{1}{2}$ . Antiderivative is $-\frac{1}{x}$ , then $-\frac{1}{2} - (-1) = \frac{1}{2}$ .

Flashcard 5: What is the integral of $f(x) = 4x$ from 1 to 3?

Answer:

Antiderivative is $2x^2$ , then $18 - 2 = 16$ .

Flashcard 6: What is the antiderivative of $f(x) = 2x^3$ ?

Answer: $\frac{x^4}{2} + C$ . Power rule integration: increase exponent, divide by new exponent.

Flashcard 7: Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

Answer: $\frac{13}{4}$ . Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.

Flashcard 8: What is the antiderivative of $f(x) = 2x^3$ ?

Answer: $\frac{x^4}{2} + C$ . Power rule integration: increase exponent, divide by new exponent.

Flashcard 9: Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.

Answer: $\frac{1}{2}$ . Antiderivative is $-\frac{1}{x}$ , then $-\frac{1}{2} - (-1) = \frac{1}{2}$ .

Flashcard 10: Find the integral of $f(x) = 1 + x$ from 0 to 2.

Answer:

Antiderivative is $x + \frac{x^2}{2}$ , evaluate at bounds.

Flashcard 11: Determine the integral of $f(x) = x^4$ from 0 to 1.

Answer: $\frac{1}{5}$ . Antiderivative is $\frac{x^5}{5}$ , then $\frac{1}{5} - 0$ .

Flashcard 12: Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.

Answer: $\frac{9}{2}$ . Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.

Flashcard 13: Identify the antiderivative given $f(x) = 2x$ .

Answer: $F(x) = x^2 + C$ . The antiderivative of $2x$ is $x^2$ plus a constant.

Flashcard 14: Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.

Answer:

Evaluate $x^3 + x^2$ at bounds 2 and 0.

Flashcard 15: Determine the integral of $f(x) = x^4$ from 0 to 1.

Answer: $\frac{1}{5}$ . Antiderivative is $\frac{x^5}{5}$ , then $\frac{1}{5} - 0$ .

Flashcard 16: Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.

Answer:

Evaluate $x^3 + x^2$ at bounds 2 and 0.

Flashcard 17: Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.

Answer:

Evaluate $\frac{x^4}{4} - x^3 + x^2$ at bounds 1 and 0.

Flashcard 18: Evaluate the integral of $f(x) = x^3$ from 0 to 2.

Answer:

Antiderivative is $\frac{x^4}{4}$ , then $\frac{16}{4} - 0 = 4$ .

Flashcard 19: Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.

Answer: $\text{ln}(2)$ . Antiderivative of $\frac{1}{x}$ is $\ln(x)$ , then $\ln(2) - \ln(1)$ .

Flashcard 20: Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.

Answer: $\text{ln}(2)$ . Antiderivative of $\frac{1}{x}$ is $\ln(x)$ , then $\ln(2) - \ln(1)$ .

Flashcard 21: What does the Fundamental Theorem of Calculus connect?

Answer: Differentiation and integration. Links the inverse operations of calculus.

Flashcard 22: Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.

Answer: $\text{ln}(4)$ . Same as $\int \frac{1}{x} dx$ , antiderivative is $\ln(x)$ .

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

Answer: Differentiation and integration. Links the inverse operations of calculus.

Flashcard 24: What is the integral of $f(x) = e^{2x}$ from 0 to 1?

Answer: $\frac{e^2 - 1}{2}$ . Antiderivative is $\frac{e^{2x}}{2}$ , evaluate at bounds.

Flashcard 25: What is the integral of $f(x) = 1$ from $a$ to $b$ ?

Answer: $b - a$ . Antiderivative of 1 is $x$ , difference of bounds gives length.

Flashcard 26: Determine the integral of $f(x) = e^x$ from 0 to 1.

Answer: $e - 1$ . Antiderivative of $e^x$ is $e^x$ , then $e - 1$ .

Flashcard 27: What is the definite integral of $f(x) = 5$ from 0 to 3?

Answer:

Integral of constant function equals constant times interval length.

Flashcard 28: What is the definite integral of $f(x) = 5$ from 0 to 3?

Answer:

Integral of constant function equals constant times interval length.

Flashcard 29: Identify the antiderivative given $f(x) = 2x$ .

Answer: $F(x) = x^2 + C$ . The antiderivative of $2x$ is $x^2$ plus a constant.

Flashcard 30: Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.

Answer:

Evaluate $\frac{x^4}{4} - x^3 + x^2$ at bounds 1 and 0.

Flashcard 31: Determine the integral of $f(x) = e^x$ from 0 to 1.

Answer: $e - 1$ . Antiderivative of $e^x$ is $e^x$ , then $e - 1$ .

Flashcard 32: Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.

Answer: $\frac{7}{3}$ . Evaluate $\frac{2x^3}{3} - \frac{3x^2}{2} + x$ at bounds.

Flashcard 33: Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$ .

Answer:

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

Flashcard 34: What is the integral of $f(x) = 1$ from $a$ to $b$ ?

Answer: $b - a$ . Antiderivative of 1 is $x$ , difference of bounds gives length.

Flashcard 35: What is the integral of $f(x) = 4x$ from 1 to 3?

Answer:

Antiderivative is $2x^2$ , then $18 - 2 = 16$ .

Flashcard 36: What is the integral of $f(x) = e^{2x}$ from 0 to 1?

Answer: $\frac{e^2 - 1}{2}$ . Antiderivative is $\frac{e^{2x}}{2}$ , evaluate at bounds.

Flashcard 37: Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.

Answer: $\frac{7}{3}$ . Evaluate $\frac{2x^3}{3} - \frac{3x^2}{2} + x$ at bounds.

Flashcard 38: Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.

Answer: $\frac{9}{2}$ . Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.

Flashcard 39: Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$ .

Answer:

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

Flashcard 40: Evaluate the integral of $f(x) = x^3$ from 0 to 2.

Answer:

Antiderivative is $\frac{x^4}{4}$ , then $\frac{16}{4} - 0 = 4$ .

AP Calculus BC Flashcards: Fundamental Theorem Of Calculus Definite Intervals

Fundamental Theorem Of Calculus Definite Intervals

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Find the integral of f(x)=4x3−x2+2f(x) = 4x^3 - x^2 + 2f(x)=4x3−x2+2 from 0 to 1.

Flashcard 2: Find the integral of f(x)=1+xf(x) = 1 + xf(x)=1+x from 0 to 2.

Flashcard 3: Calculate the integral of f(x)=x−1f(x) = x^{-1}f(x)=x−1 from 1 to 4.

Flashcard 4: Calculate the integral of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​ from 1 to 2.

Flashcard 5: What is the integral of f(x)=4xf(x) = 4xf(x)=4x from 1 to 3?

Flashcard 6: What is the antiderivative of f(x)=2x3f(x) = 2x^3f(x)=2x3?

Flashcard 7: Find the integral of f(x)=4x3−x2+2f(x) = 4x^3 - x^2 + 2f(x)=4x3−x2+2 from 0 to 1.

Flashcard 8: What is the antiderivative of f(x)=2x3f(x) = 2x^3f(x)=2x3?

Flashcard 9: Calculate the integral of f(x)=1x2f(x) = \frac{1}{x^2}f(x)=x21​ from 1 to 2.

Flashcard 10: Find the integral of f(x)=1+xf(x) = 1 + xf(x)=1+x from 0 to 2.

Flashcard 11: Determine the integral of f(x)=x4f(x) = x^4f(x)=x4 from 0 to 1.

Flashcard 12: Find the value of the integral of f(x)=x2−xf(x) = x^2 - xf(x)=x2−x from 0 to 3.

Flashcard 13: Identify the antiderivative given f(x)=2xf(x) = 2xf(x)=2x.

Flashcard 14: Find the integral of f(x)=3x2+2xf(x) = 3x^2 + 2xf(x)=3x2+2x from 0 to 2.

Flashcard 15: Determine the integral of f(x)=x4f(x) = x^4f(x)=x4 from 0 to 1.

Flashcard 16: Find the integral of f(x)=3x2+2xf(x) = 3x^2 + 2xf(x)=3x2+2x from 0 to 2.

Flashcard 17: Evaluate the integral of f(x)=x3−3x2+2xf(x) = x^3 - 3x^2 + 2xf(x)=x3−3x2+2x from 0 to 1.

Flashcard 18: Evaluate the integral of f(x)=x3f(x) = x^3f(x)=x3 from 0 to 2.

Flashcard 19: Evaluate the integral of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from 1 to 2.

Flashcard 20: Evaluate the integral of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from 1 to 2.

Flashcard 21: What does the Fundamental Theorem of Calculus connect?

Flashcard 22: Calculate the integral of f(x)=x−1f(x) = x^{-1}f(x)=x−1 from 1 to 4.

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

Flashcard 24: What is the integral of f(x)=e2xf(x) = e^{2x}f(x)=e2x from 0 to 1?

Flashcard 25: What is the integral of f(x)=1f(x) = 1f(x)=1 from aaa to bbb?

Flashcard 26: Determine the integral of f(x)=exf(x) = e^xf(x)=ex from 0 to 1.

Flashcard 27: What is the definite integral of f(x)=5f(x) = 5f(x)=5 from 0 to 3?

Flashcard 28: What is the definite integral of f(x)=5f(x) = 5f(x)=5 from 0 to 3?

Flashcard 29: Identify the antiderivative given f(x)=2xf(x) = 2xf(x)=2x.

Flashcard 30: Evaluate the integral of f(x)=x3−3x2+2xf(x) = x^3 - 3x^2 + 2xf(x)=x3−3x2+2x from 0 to 1.

Flashcard 31: Determine the integral of f(x)=exf(x) = e^xf(x)=ex from 0 to 1.

Flashcard 32: Find the integral of f(x)=2x2−3x+1f(x) = 2x^2 - 3x + 1f(x)=2x2−3x+1 from 1 to 2.

Flashcard 33: Calculate the integral of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from 1 to eee.

Flashcard 34: What is the integral of f(x)=1f(x) = 1f(x)=1 from aaa to bbb?

Flashcard 35: What is the integral of f(x)=4xf(x) = 4xf(x)=4x from 1 to 3?

Flashcard 36: What is the integral of f(x)=e2xf(x) = e^{2x}f(x)=e2x from 0 to 1?

Flashcard 37: Find the integral of f(x)=2x2−3x+1f(x) = 2x^2 - 3x + 1f(x)=2x2−3x+1 from 1 to 2.

Flashcard 38: Find the value of the integral of f(x)=x2−xf(x) = x^2 - xf(x)=x2−x from 0 to 3.

Flashcard 39: Calculate the integral of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ from 1 to eee.

Flashcard 40: Evaluate the integral of f(x)=x3f(x) = x^3f(x)=x3 from 0 to 2.

Flashcard 1: Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

Flashcard 2: Find the integral of $f(x) = 1 + x$ from 0 to 2.

Flashcard 3: Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.

Flashcard 4: Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.

Flashcard 5: What is the integral of $f(x) = 4x$ from 1 to 3?

Flashcard 6: What is the antiderivative of $f(x) = 2x^3$ ?

Flashcard 7: Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.

Flashcard 8: What is the antiderivative of $f(x) = 2x^3$ ?

Flashcard 9: Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.

Flashcard 10: Find the integral of $f(x) = 1 + x$ from 0 to 2.

Flashcard 11: Determine the integral of $f(x) = x^4$ from 0 to 1.

Flashcard 12: Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.

Flashcard 13: Identify the antiderivative given $f(x) = 2x$ .

Flashcard 14: Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.

Flashcard 15: Determine the integral of $f(x) = x^4$ from 0 to 1.

Flashcard 16: Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.

Flashcard 17: Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.

Flashcard 18: Evaluate the integral of $f(x) = x^3$ from 0 to 2.

Flashcard 19: Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.

Flashcard 20: Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.

Flashcard 22: Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.

Flashcard 24: What is the integral of $f(x) = e^{2x}$ from 0 to 1?

Flashcard 25: What is the integral of $f(x) = 1$ from $a$ to $b$ ?

Flashcard 26: Determine the integral of $f(x) = e^x$ from 0 to 1.

Flashcard 27: What is the definite integral of $f(x) = 5$ from 0 to 3?

Flashcard 28: What is the definite integral of $f(x) = 5$ from 0 to 3?

Flashcard 29: Identify the antiderivative given $f(x) = 2x$ .

Flashcard 30: Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.

Flashcard 31: Determine the integral of $f(x) = e^x$ from 0 to 1.

Flashcard 32: Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.

Flashcard 33: Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$ .

Flashcard 34: What is the integral of $f(x) = 1$ from $a$ to $b$ ?

Flashcard 35: What is the integral of $f(x) = 4x$ from 1 to 3?

Flashcard 36: What is the integral of $f(x) = e^{2x}$ from 0 to 1?

Flashcard 37: Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.

Flashcard 38: Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.

Flashcard 39: Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$ .

Flashcard 40: Evaluate the integral of $f(x) = x^3$ from 0 to 2.