1. 5.3 Definite Integrals

2. Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles

3. Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles Let’s try 4 rectangles…  1.97

4. Integration Symbol lower limit of integration upper limit of integration integrand variable of integration So what do all of these symbols mean? Remember from last time:

5. Don’t forget that we are still finding the area under the curve. Now, how to evaluate the Definite Integral… Summation Symbol Rectangle Height Rectangle Base

6. And therefore that… Let F(x) be a function such that F'(x) = f(x) In other words, f is the derivative of F and F is the anti-derivative of f. From this we can also infer that… This will be useful…

7. xx Now let’s get the notation straight: …and therefore

8. xx And if we remove the limit (making it an approximation) Now let’s get the notation straight: and multiply both sides by  x, we get

9.    Since we are basing this on the left hand method, we stop at k – 1 Now let’s write out each of the rectangles… k = 1 k = 2 k = 3   

10. Now when we we add up all of the rectangles, what happens on the right side of the approximation?

11. The Definite Integral

12. Example: Find the area under the curve from x = 1 to x = 2. Area under the curve from x = 1 to x = 2. There can be a slight difference between finding the total area under the curve and finding the integral as we shall see next…

13. Example: Find the area between the x -axis and the curve from to. pos. neg. AHA! But here we need to remember that area under the x-axis will be negative. So let’s be careful what we are looking for.

14. Example: Find the definite integral of from to. pos. neg. If we simply wanted the integral from 0 to then we would just take one integral. = –1 – 0 =–1 …and notice that the lower area is –2 and the upper area is 1. so the integral is the sum of the two numbers not the sum of the actual areas.

15. Page 269 gives rules for working with integrals, the most important of which are: 2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted.

16. 5. Intervals can be added (or subtracted.) 4. Integrals can be added and subtracted.