1. 5.2 Definite Integral Tues Nov 15
Do Now
Find the area under f(x) = 3 – x in the interval [0,3] using 3 leftendpt rectangles
Find the actual area

2. The Definite Integral
When working with Riemann Sums, the width of each rectangle does not have to be uniform, but the idea still persists:
Exact area under curve =

3. The Definite Integral
The definite integral of f(x) over [a,b], denoted by the integral sign, is the limit of Riemann sums:
When this limit, exists, f(x) is integrable over [a,b]

4. Theorem
If f(x) is continuous on [a, b], or if f(x) is continuous with at most finitely many jump discontinuities, then f(x) is integrable over [a, b]

5. Area above and below x-axis
Any area can be bounded by the x-axis and the function:
If the area is above the x-axis, then it is considered positive
If the area is below the x-axis, then it is considered negative

6. Ex
Calculate

7. Integral of a Constant
For any constant C,

8. Properties of Integrals

9. Reversing the Limits of Integration
If we reverse the limits of integration,

10. Additivity for Adjacent Intervals
Let , and assume that f(x) is integrable.
Then:
This is useful for absolute value or piecewise functions

11. Ex
Evaluate the integral

12. Closure If
HW: p.307 #5, 11, 13, 19, 25, 31, 33, 43, 45, 47, 55, 61, 65, 71

13. 5.2 Review / Practice FRQ Wed Nov 18
Do Now
Evaluate

14. HW Review

15. Practice FRQ
Work for 15 min