Approximating Areas With Riemann Sums - AP Calculus BC

1. Calculate $M_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.

   $M_4 = 8$. $\triangle x = 1$; sum $f(0.5) + f(1.5) + f(2.5) + f(3.5) = 0.5 + 1.5 + 2.5 + 3.5 = 8$.

2. Which Riemann sum uses midpoints of intervals?

   Midpoint Riemann sum. Uses center point of each subinterval for height calculation.

3. State the formula for the midpoint Riemann sum.

   $M_n = \text{sum of } f(m_i)\times \text{width}$, using midpoints. Each rectangle height uses function value at interval's center.

4. What is the general form of a Riemann sum?

   $\text{sum of } f(x_i^*) \times \triangle x$. Standard notation for all Riemann sum variations.

5. Calculate $L_2$ for $f(x)=x^3$ on $[1,3]$ with $n=2$.

   $L_2 = 9$. $\triangle x = 1$; sum $f(1) + f(2) = 1 + 8 = 9$.

6. How does a left Riemann sum differ from a right Riemann sum?

   Uses left vs. right endpoints of subintervals. Different sampling points within each subinterval affect approximation.

7. What is the midpoint of the interval $[2,6]$?

   $m = 4$. Average of interval endpoints: $(2+6)/2 = 4$.

8. How is a Riemann sum related to the definite integral?

   It approximates the integral as $n \to \text{infinity}$. Riemann sum approaches integral value as partition size approaches zero.

9. Find $\triangle x$ for $n=4$ on $[0,8]$.

   $\triangle x = 2$. $(8-0)/4 = 2$ for the given interval and partitions.

10. Calculate $R_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.

    $R_4 = 10$. $\triangle x = 1$; sum $f(1) + f(2) + f(3) + f(4) = 1 + 2 + 3 + 4 = 10$.

11. What is a Riemann sum?

    A method for approximating the total area under a curve. Divides interval into rectangles to estimate area under curve.

12. Identify the type of Riemann sum that tends to overestimate.

    Right Riemann sum (if $f(x)$ is increasing). Right endpoints sample higher function values on increasing functions.

13. Calculate $L_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.

    $L_4 = 6$. $\triangle x = 1$; sum $f(0) + f(1) + f(2) + f(3) = 0 + 1 + 2 + 3 = 6$.

14. Calculate $M_2$ for $f(x)=x^2$ on $[0,2]$ with $n=2$.

    $M_2 = 2.5$. $\triangle x = 1$; midpoints $0.5, 1.5$ give $(0.5)^2 + (1.5)^2 = 2.5$.

15. How is the partition width $\triangle x$ calculated?

    $\triangle x = \frac{b-a}{n}$, where $[a,b]$ is the interval. Interval length divided by number of subintervals.

16. What is necessary for a Riemann sum to equal the exact area?

    $n \to \text{infinity}$ and $f(x)$ continuous. Infinite partitions with continuous functions guarantee convergence to exact area.

17. Identify the type of Riemann sum that tends to underestimate.

    Left Riemann sum (if $f(x)$ is increasing). Left endpoints sample lower function values on increasing functions.

18. Identify the midpoint for the interval $[3,5]$.

    $m = 4$. Average of interval endpoints: $(3+5)/2 = 4$.

19. State the formula for the left Riemann sum.

    $L_n = \text{sum of } f(x_i^*)\times \text{width}$, using left endpoints. Each rectangle height uses function value at left edge of interval.

20. What is the role of $n$ in a Riemann sum?

    Number of subintervals for the partition. Controls precision by determining how many subintervals are used.

21. Identify the partition width for $n=4$ on $[1,5]$.

    $\triangle x = 1$. $(5-1)/4 = 1$ for the given interval and partitions.

22. What is the effect of increasing $n$ on Riemann sums?

    Increases accuracy of the approximation. More rectangles provide finer approximation of curved area.

23. What is the trapezoidal rule?

    An approximation method using trapezoids to estimate area. Connects consecutive points with straight lines to form trapezoids.

24. What does $\triangle x$ represent in Riemann sums?

    Width of each subinterval. Size of each rectangular subdivision in the partition.

25. State the formula for the trapezoidal rule.

    $T_n = \frac{b-a}{2n} [f(x_0) + 2\text{sum of } f(x_i) + f(x_n)]$. Averages left and right endpoints, summing trapezoid areas.

26. What is the effect of decreasing $n$ on approximation?

    Decreases accuracy of the approximation. Fewer rectangles provide coarser approximation of area.

27. Calculate $\triangle x$ for $n=5$ on $[2,12]$.

    $\triangle x = 2$. $(12-2)/5 = 2$ for the given interval and partitions.

28. What is the primary use of Riemann sums?

    To approximate the integral of a function. Estimates area under curves when exact integration is difficult.

29. How does the trapezoidal rule improve accuracy?

    Uses trapezoids instead of rectangles. Trapezoids better approximate curved regions than rectangles.

30. State the formula for the right Riemann sum.

    $R_n = \text{sum of } f(x_i^*)\times \text{width}$, using right endpoints. Each rectangle height uses function value at right edge of interval.

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