Riemann Sums and Notation - AP Calculus BC

1. What is $\int_{0}^{1} x^2 \, dx$?

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

2. What is the integral of $e^x$ from $0$ to $1$?

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

3. What is the general formula for a Riemann sum?

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

4. What is the symbol for the integral sign in calculus notation?

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

5. What is the integral of $x$ with respect to $x$ over $[0, 1]$?

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

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

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

7. Which rule uses the average of left and right Riemann sums?

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

8. Which Riemann sum uses the right endpoint of subintervals?

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

9. What is the standard notation for summation?

   $\sum_{i=m}^{n} a_i$. Sums terms from index $m$ to $n$ using variable $a_i$.

10. What is the trapezoidal rule for approximating integrals?

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

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

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

12. What is the limit definition of a definite integral?

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

13. Evaluate $\int_{0}^{4} (3t^2) \, dt$.

    64. Antiderivative is $t^3$; $4^3 - 0^3 = 64$.

14. What is the purpose of a Riemann sum?

    Approximate the area under a curve. Uses rectangles to estimate area under curves numerically.

15. Write the expression for a left Riemann sum.

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

16. What is the formula for the left Riemann sum?

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

17. What does $x_i$ represent in summation notation?

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

18. What is the integral of $\cos(x)$ from $0$ to $\pi$?

    0. Antiderivative $\sin(x)$ gives $\sin(\pi) - \sin(0) = 0$.

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

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

20. What is $\int_{1}^{3} (2x + 1) \, dx$?

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

21. Identify $\int_{a}^{b} c \, dx$, where $c$ is a constant.

    $c(b-a)$. Constant times the interval length $(b-a)$.

22. State the definition of a definite integral.

    The limit of a Riemann sum as $n \to \infty$. As partitions get finer, Riemann sums approach the exact area.

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

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

24. What is the geometric interpretation of a definite integral?

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

25. What does the symbol $\Sigma$ represent in mathematics?

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

26. Which method uses rectangles to approximate areas under curves?

    Riemann sum. Divides interval into rectangles for area approximation.

27. Which theorem connects derivatives and definite integrals?

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

28. What does a definite integral compute?

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

29. Identify the integral of a constant $c$ over $[a, b]$.

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

30. Find $\int_{1}^{2} x^3 \, dx$.

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

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

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

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

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

33. What does $\Delta x$ represent in Riemann sums?

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

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

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

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

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

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

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

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