1. x01234 f(x)f(x)94101

2. AP Calculus Unit 5 Day 5 Integral Definition and intro to FTC

3. Riemann Sums… Thus far, we have used rectangles and trapezoids to APPROXIMATE area between curves and the x-axis. It would be better if we could be more accurate in our approximations. Riemann Sum -- from Wolfram MathWorld

4. k th rectangle Rectangles extending from the x-axis to intersect the curve at the points (c k,f(c k )) Height of k th rectangle (c k,f(c k )) Width of k th rectangle=  x k (c n,f(c n )) xkxk x k-1 ckck cncn x n-1 x n =b x 0 =a c1c1 c2c2 x1x1 x2x2 (c 1,f(c 1 )) (c 2,f(c 2 )) Notes Here: 1. Rectangles are used to approximate 2. The area of any specific rectangle will be: where is the height and is the width.

5. k th rectangle Rectangles extending form from the x-axis to intersect the curve at the points (c k,f(c k )) Height of k th rectangle (c k,f(c k )) Width of k th rectangle=  x k (c n,f(c n )) xkxk x k-1 ckck cncn x n-1 x n =b x 0 =a c1c1 c2c2 x1x1 x2x2 (c 1,f(c 1 )) (c 2,f(c 2 )) Notes Here: 3. The product will be positive or negative based on the value of 4. The sum of these areas can be written as:

6. k th rectangle Rectangles extending form from the x-axis to intersect the curve at the points (c k,f(c k )) Height of k th rectangle (c k,f(c k )) Width of k th rectangle=  x k (c n,f(c n )) xkxk x k-1 ckck cncn x n-1 x n =b x 0 =a c1c1 c2c2 x1x1 x2x2 (c 1,f(c 1 )) (c 2,f(c 2 )) Notes Here: 5. As we increase the number of rectangles the approximation becomes more accurate. This can be written using the notation that is on the next slide.

7. If we use an infinite number of partitions… Width of each partition Height of partitions Add up the areas of each partition The number of partitions is increasing to infinity

8. Integral Notation And since represents the EXACT amount We can state that

9. Two ways to view the limit Number of partitions goes to infinity Size of partitions goes to zero

10. Both equal the integral

11. Formal Definition : Let f be a function on a closed interval [a,b], let the numbers c k be chosen arbitrarily in the subintervals [x k-1, x k ]. If there exists a number I such that no matter how P and c k ’s are chosen, Then f is integrable on [a,b] and I is the definite integral of f over [a,b].

12. Examples: Write as an integral:

13. Examples: Write as an integral:

14. Examples: Write as an integral:

15. Explore Properties

16. Explain why this makes sense based on your knowledge of what an integral represents.