1. SECTION 4.2: AREA AP Calculus BC

2. LEARNING TARGETS: Use Sigma Notation to evaluate a sum Apply area formulas from geometry to determine the area under a curve. Approximate the area of a plane region using LRAM, MRAM, and RRAM.

3. RECTANGULAR APPROXIMATION METHODS Estimates the area under a curve using rectangular areas. MRAM—Uses the midpoint of the subinterval to determine the height of the rectangle. LRAM—Uses the left endpoint of the subinterval to determine the height of the rectangle. RRAM—Uses the right endpoint of the subinterval to determine the height of the rectangle.  RRAM  LRAM

4. SIGMA NOTATION Examples:

5. EVALUATING A SUM (PAGE 255)

6. APPROXIMATING THE AREA

8. EXAMPLE 2:

9. UPPER AND LOWER SUMS Inscribed Rectangles = lying inside the subregion The sum of the areas of the inscribed regions is the lower sum The sum of the areas of the inscribed regions is the lower sum Notation: Notation: Circumscribed Rectangles = extending outside of the subregion. The sum of the areas of the circumscribed regions is the upper sum The sum of the areas of the circumscribed regions is the upper sum Notation: Notation:

10. LIMIT DEFINITION OF LOWER AND UPPER SUMS

11. EXAMPLE 3:

12. EXAMPLE 4:

13. MIDPOINT RULE As an alternative to using Upper and Lower Sums, the midpoint of the interval may lead to good or even better approximations.

14. EXAMPLE 5 – MIDPOINT RULE: As an alternative to using Upper and Lower Sums, the midpoint of the interval may lead to good or even better approximations.

15. HOMEWORK: pg. 263–264: #9, 11, 21 – 27 odd, 33, 35, 45, 49, 53, 55, 57, 61, 64