AP Calculus BC Flashcards: Riemann Sums And Notation

What this deck covers

This deck focuses on Riemann Sums And Notation, 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: What is the symbol for the integral sign in calculus notation?

Answer: $\int$. Elongated S symbol representing integration operation.

Flashcard 2: Which method uses rectangles to approximate areas under curves?

Answer: Riemann sum. Divides interval into rectangles for area approximation.

Flashcard 3: What is the trapezoidal rule for approximating integrals?

Answer: $\frac{b-a}{2n} \sum_{i=1}^{n} (f(x_{i-1}) + f(x_i))$. Averages left and right endpoint values for better approximation.

Flashcard 4: Find the definite integral: $\int_{0}^{2} (3x^2 + 2x) \, dx$.

Answer:

1. Antiderivative $x^3 + x^2$ gives $(8+4) - (0+0) = 12$.

Flashcard 5: Write the notation for a definite integral from $a$ to $b$ of $f(x)$.

Answer: $\int_{a}^{b} f(x) \, dx$. Standard integral notation with limits of integration.

Flashcard 6: What is the integral of $e^x$ from $0$ to $1$?

Answer: $e - 1$. Antiderivative $e^x$ gives $e^1 - e^0 = e - 1$.

Flashcard 7: What is the general formula for a Riemann sum?

Answer: $\text{Sum} = \frac{b-a}{n} \times \text{sum of } f(x_i)$. Multiplies interval width by sum of function values at sample points.

Flashcard 8: What does a definite integral compute?

Answer: Net area between the curve and the x-axis. Positive above x-axis, negative below, giving signed area.

Flashcard 9: Which Riemann sum uses the right endpoint of subintervals?

Answer: Right Riemann sum. Uses right boundary of each subinterval for height.

Flashcard 10: Evaluate $\int_{0}^{1} (4x^3) \, dx$.

Answer: $1$. Antiderivative is $x^4$; $1^4 - 0^4 = 1$.

Flashcard 11: Find $\int_{1}^{2} x^3 \, dx$.

Answer: $\frac{15}{4}$. Antiderivative $\frac{x^4}{4}$ gives $\frac{16}{4} - \frac{1}{4} = \frac{15}{4}$.

Flashcard 12: What is the formula for the left Riemann sum?

Answer: $\sum_{i=0}^{n-1} f(x_i) \Delta x$. Standard formula using left endpoints for approximation.

Flashcard 13: What is $\int_{1}^{3} (2x + 1) \, dx$?

Answer:

1. Antiderivative is $x^2 + x$; evaluating gives $(9+3)-(1+1) = 8$.

Flashcard 14: Write the expression for a left Riemann sum.

Answer: $\text{Sum} = \sum_{i=0}^{n-1} f(x_i) \Delta x$. Uses left endpoints of subintervals for rectangle heights.

Flashcard 15: Identify the error: $\frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x)$.

Answer: Correct: $\frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x)$. No error; this correctly states the Fundamental Theorem of Calculus.

Flashcard 16: Express the right Riemann sum for $f(x)$ on $[a, b]$.

Answer: $\text{Sum} = \sum_{i=1}^{n} f(x_i) \Delta x$. Uses right endpoints of subintervals for function evaluation.

Flashcard 17: What does $\Delta x$ represent in Riemann sums?

Answer: Width of each subinterval. Calculated as $(b-a)/n$ for uniform partitions.

Flashcard 18: What is the geometric interpretation of a definite integral?

Answer: Area under the curve $f(x)$ from $x = a$ to $x = b$. Represents the signed area between function and x-axis.

Flashcard 19: What does the symbol $\Sigma$ represent in mathematics?

Answer: Summation. Greek letter sigma denotes sum of terms in a sequence.

Flashcard 20: Find $\int_{0}^{\pi} \sin(x) \, dx$.

Answer:

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

Flashcard 21: What does $x_i$ represent in summation notation?

Answer: A sample point in the $i^{th}$ subinterval. Chosen point within each partition for function evaluation.

Flashcard 22: Determine the value of $\int_{2}^{5} 3 \, dx$.

Answer:

1. Constant 3 integrated over interval length $5-2 = 3$.

Flashcard 23: Which rule uses the average of left and right Riemann sums?

Answer: Trapezoidal rule. Combines left and right sums for improved accuracy.

Flashcard 24: Identify the integral of a constant $c$ over $[a, b]$.

Answer: $c(b-a)$. Integral of constant equals constant times interval width.

Flashcard 25: Find the midpoint Riemann sum for $f(x)$ on $[a, b]$.

Answer: $\text{Sum} = \sum_{i=0}^{n-1} f(\frac{x_i + x_{i+1}}{2}) \Delta x$. Evaluates function at midpoint of each subinterval.

Flashcard 26: What is $\int_{0}^{1} x^2 \, dx$?

Answer: $\frac{1}{3}$. Antiderivative of $x^2$ is $\frac{x^3}{3}$, evaluated from 0 to 1.

Flashcard 27: Which theorem connects derivatives and definite integrals?

Answer: Fundamental Theorem of Calculus. Links antiderivatives to definite integral evaluation.

Flashcard 28: What is the limit definition of a definite integral?

Answer: $\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$. Riemann sum limit as partition size approaches zero.

Flashcard 29: What does $\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x$ represent?

Answer: Definite integral of $f(x)$ over $[a, b]$. Limit of Riemann sums defines the definite integral.

Flashcard 30: What is the integral of $x$ with respect to $x$ over $[0, 1]$?

Answer: $\frac{1}{2}$. Antiderivative $\frac{x^2}{2}$ evaluated from 0 to 1 gives $\frac{1}{2}$.