1. 4.3 Day 1 Exit Ticket for Feedback
A particle starts at 𝑥=−1 when 𝑡=0 and moves along the x-axis with velocity 𝑣 𝑡 =2𝑡+3 for time 𝑡≥0. Where is the particle at 𝑡=3?

2. Section 4.3 Day 2 Riemann Sums and Definite Integrals
AP Calculus AB

3. Learning Targets Define Riemann Sums
Conceptually connect approximation and limits
Evaluate left hand, right hand and midpoint Riemann Sums of equal and unequal lengths from graphs & tables
Evaluate approximations using the trapezoidal rule
Define a definite integral
Evaluate a definite integral geometrically and with a calculator
Define an integral in terms of area
Apply properties of a definite integral

4. Set up Foldable
Cover: 4.3 Day 2 – Day 3: Riemann Sums Conceptual Riemann Sum
Left Hand Riemann Sum
Right Hand Riemann Sum
Midpoint Riemann Sum
Trapezoidal Riemann Sum

5. Conceptual Riemann Sums
In your groups, briefly discuss the situation where the approximation of the area under a curve with rectangles can be the exact area under the curve.
As the number of rectangles approximating the area under the curve gets closer to infinity, the area under the curve becomes closer to the exact area. The number of rectangles as a limit to infinity will produce the exact area under the curve

6. Summary
lim 𝑛→∞ 𝑘=1 𝑛 𝑓 𝑐 𝑘 ∆𝑥 𝑓 𝑐 𝑘 ∆𝑥 represents the areas of the rectangles. (height times base) The summation notation represents adding all of the area of the rectangles together. The limit notation represents having an infinite number of rectangles which provides us with the exact area under the curve.

7. Left Hand Riemann Sums
Definition: Use the left height of the rectangle to determine the area of the rectangle. Equal Subdivisions: 𝑏−𝑎 𝑛 on the interval [𝑎,𝑏] with 𝑛 rectangles Unequal Subdivisions: Calculate the area of each rectangle. Then, add them all together

8. Left Hand Riemann Sums
Example 1 – Equal Subdivisions Find the LH Riemann approximation for the function 𝑓(𝑥)= 𝑥 2 over [0,2] with 4 partitions/rectangles. Base: 1 2 Area: 1 2 [𝑓 0 +𝑓 1 2 +𝑓 1 +𝑓 3 2 ] Under or over approximation?

9. Left Hand Riemann Sums
Example 2 – Unequal Subdivisions Use a LH Riemann approximation for the following function with 4 rectangles =52 𝑥 2 3 7 9
𝑓(𝑥) 6 8

10. Right Hand Riemann Sums
Definition: Use the right height of the rectangle to determine the area of the rectangle. Equal Subdivisions: 𝑏−𝑎 𝑛 on the interval [𝑎,𝑏] with 𝑛 rectangles Unequal Subdivisions: Calculate the area of each rectangle. Then, add them all together

11. Right Hand Riemann Sums
Example 1 – Equal Subdivisions Find the RH Riemann approximation for the function 𝑓(𝑥)= 𝑥 2 over [1,3] with 6 partitions/rectangles. Base: 2 6 = 1 3 Area: 1 3 [𝑓 4 3 +𝑓 5 3 +𝑓 2 +𝑓 7 3 +𝑓 𝑓(3)] Under or over approximation?

12. Right Hand Riemann Sums
Example 2 – Unequal Subdivisions Use a RH Riemann approximation for the following function with 4 rectangles =59 𝑥 2 3 7 9
𝑓(𝑥) 6 8

13. Midpoint Riemann Sums
Definition: Use the middle height of the rectangle to determine the area of the rectangle. Equal Subdivisions: 𝑏−𝑎 𝑛 on the interval [𝑎,𝑏] with 𝑛 rectangles Unequal Subdivisions: Calculate the area of each rectangle. Then, add them all together

14. Midpoint Riemann Sums
Example 1 – Equal Subdivisions Find the MP Riemann approximation for the function 𝑓(𝑥)= 𝑥 2 over [0,2] with 4 partitions/rectangles. Base: 1 2 Area: 1 2 [𝑓 1 4 +𝑓 3 4 +𝑓 5 4 +𝑓 7 4 ]

15. Midpoint Riemann Sums
Example 2 – Equal Subdivisions Use a MP Riemann approximation for the following function with 4 partitions/ rectangles =229 𝒙 5 10 15 20 25 30 35 40
𝑓(𝑥)
7.0
9.2
9.5
4.5
2.4
4.3
7.3

16. Trapezoidal Riemann Sum
Definition: Use trapezoids with the “height” of the rectangles as the bases of the trapezoid Equal Subdivisions: 𝑏−𝑎 𝑛 1 2 𝑓 𝑥 0 +2𝑓 𝑥 1 +2𝑓 𝑥 2 +…+𝑓 𝑥 𝑛 Unequal Subdivisions: Find the area of each trapezoid 𝑨= 𝒃 𝟏 + 𝒃 𝟐 𝟐 ∙𝒉

17. Trapezoidal Riemann Sum
Example 1 – Equal Subdivisions Find the trapezoidal approximation for the function 𝑓(𝑥)= 𝑥 2 over [0,2] with 4 partitions. Area: 𝑓 0 +𝑓 ∙ 𝑓 1 2 +𝑓 1 2 ∙ 𝑓 1 +𝑓 ∙ 𝑓 3 2 +𝑓 2 2 ∙ 1 2

18. Trapezoidal Riemann Sum
Example 2 – Unequal Subdivisions Use a trapezoidal approximation for the following function with 4 partitions ∙ ∙ ∙ ∙1 𝒙 2 5 9 10
𝑓(𝑥) 66 60 52 44 43

19. Homework
4.3 Day 2 – Day 3 Homework Worksheet