1. MA132 Final exam Review

2. 6.1 Area between curves Partition into rectangles! Area of a rectangle is A = height*base Add those up ! (Think: Reimann Sum) For the height, think “top – bottom”

3. 6.2 Volumes by slicing Given a region bounded by curves Rotate that region about the x-axis, y-axis, or a horizontal or vertical line Generate a solid of revolution Partition into disks

4. 6.2 Volume by slicing Consider a slice perpendicular to the axis of rotation Consider a slice perpendicular to the line of rotation Label the thickness This slice will be a disk or a washer We can find the volume of those! Consider a partition and add them up (Think Reimann sum)

5. Disks and Washer roro riri r o is the distance from the line of rotation to the outer curve. r i is the distance from the line of rotation to the inner curve.

6. roro riri

7. Idea works for functions of y, too

8. 6.3 Volume by Shells Consider a rectangle parallel to the line of rotation Label the thickness Rotating that rectangle around leads to a cylindrical shell We can find the volume of those! Consider a partition and add them up (Think Reimann sum) A cool movie

9. Setting up the integral Another cool movie Another cool movie

10. Shell Hints Draw the reference rectangle and a shell Label everything! The radius is just the distance from the line of rotation to the ‘reference rectangle’ ALWAYS think in terms of distances Radius here is just x x=d d xx Radius here is (d – x)

11. Chapter 7: Techniques of Integration Integration by Parts Trig Integrals (i.e. using identities for clever u-sub) Trig Substitution Partial Fractions Improper Integrals

12. 7.1 Parts: for handling products of functions Choose u so that differentiating leads to an easier function Choose dv so that you know how to integrate it! Be aware of boomerangs in life (not on the final) Careful:

13. 7.2 Trig Integrals Use a trig identity to find an integral with a clever u-substituion! Examine what the possibilities for ‘du’ are and then use the identities to get everything else in terms of ‘u’

14. 7.4 Trig Substitution Use Pythagorean Identities Use a change of variables Rewrite everything in terms of trig functions –May have to apply more trig identities Change back to original variable! –May need to draw a right triangle!

15. 7.3 Trig Sub Use Algebra to rewrite in this form

16. Trig sub pitfalls Do NOT use the same variable when you make a ‘change of variables’ –EX. Let x=sin(x) Do NOT forget to include ‘dx’ when you rewrite your integral Do NOT forget to change BACK to the original variable –May involve setting up a right triangle –You may need to use sin(2x)=2sin(x)cos(x)

17. 7.4 Partial Fractions IDEA: We do not know how to integrate But we do know how to integrate These are equal! We just need algebra!

18. Undo the process of getting a common denominator Must be proper rational function  Degree of numerator < degree of denominator FACTOR  product of linear terms and irreducible quadratic terms FORM FIND

19. Forming the PFD: depends on the factored Q(x) Q(x) includes distinct linear terms, include one of these for each one! Q(x) includes some repeated linear terms, include one term for each—with powers up to the repeated value

20. Forming the PFD: depends on the factored Q(x) Q(x) includes irreducible quadratics Q(x) includes repeated irreducible quadratics

21. Forming the PFD: depends on the factored Q(x) Or a combination of all those! Example:

22. 7.8 Improper Integrals Two Types: Infinite bounds Singularity between the bounds Singularity at x=a Integrating to infinity

23. Plan of attack Rewrite using a dummy variable and in terms of a limit Integrate! Evaluate the limit of the result Analyze the result –A finite number: integral converges –Otherwise: integral diverges These involve Integration AND limits

24. Differential Equations An equation involving an unknown function and some of its derivatives We looked at separation of variables (9.3) Applications (9.4) –Growth/population models –Newton’s law of cooling

25. 9.3 Separable DEs

26. Separable DEs Remember the constant of integration Initial value problems –Given an initial condition y(x 0 )=y 0 –Use to define the value of C Implicit solution vs. Explicit solution

27. 9.4 Applications The rate of growth is proportional to the population size The rate of cooling is proportional to the temperature difference between the object and its surroundings These are separable differential equations

28. Sequences and Series 11.1 Sequences 11.2 Series 11.4-11.6 Series tests (no 11.3) 11.8 Power series 11.9 functions of power series 11.10 MacLaurin and Taylor series

29. 11.1 Sequences Some ideas Don’t forget everything you know about limits! Only apply L’Hopital’s rule to continuous functions of x Do NOT apply series tests!

30. Series Know which tests apply to positive series and ALL conditions for each test Absolute convergence means converges Absolute convergence implies convergence Conditional convergence means converges BUT does NOT

31. Power Series Make repeated use of the ratio test! For what values of x does the series converge

32. Idea Given Apply ratio test: This limit should include |x-a| Unless the limit is 0 or infinity We set L<1 because That is when the Ratio Test yields convergence Then use algebra to express This as |x-a|<r

33. Functions as Power Series

34. Taylor and MacLaurin Series KNOW the MacLaurin series for –sin(x) – cos(x) – e x