Understanding Area, Riemann Sums, and Definite Integrals

Antiderivatives are utilized to find area under curves, where Riemann Sums are employed for approximations. The process involves dividing intervals into rectangles for both approximate and exact area calculations. Definite integrals provide specific, finite values representing total displacement, with properties and rules like the Trapezoid Rule offering further insights. Various homework exercises are suggested for practice.

• Antiderivatives
• Riemann Sums
• Definite Integrals
• Trapezoid Rule
• Homework

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1. 4.2/4.3 Area, Riemann Sums & Definite Integrals

2. *Antiderivatives can be used to find area under a curve. Some antiderivatives can t be found directly so we must approximate them using Riemann Sums.

3. Riemann Sums We find the area under a curve, both approximate and exact, by dividing the interval into rectangles. *more rect. more accurate approx. *as # rectangles , we approach the exact area

4. On : Width of a rectangle: n= # of rectangles *can use left endpoints, right endpoints, or midpoints

5. Definite Integral: : *specific, finite value, no C *represents total displacement *must be continuous on [a, b]

6. Properties

7. Trapezoid Rule: w/ n subintervals *width of interval: *subintervals must be equal

8. Homework Day 1, p. 299: 29 (use left, right, and midpoints with 4 rectangles), 33, 35 Day 2, p. 312: 37-47 odd, 49, 57, 59, 110-113 all