(MTH 250)

1. (MTH 250) Calculus Lecture 21

2. Previous Lecture’s Summary • Trigonometricintegrals • Trigonometric substitutions • Partial fractions

3. Today’s Lecture • Recalls • Improperintegrals • Introduction to vectors • Dot productof vectors • Cross product of vectors

4. Recalls When evaluating integrals of the form • If the power of sine is odd save one sine factor. • Use sin2x = 1 - cos2x to express the remaining factors in terms of cosine: • Then, substitute . • If the powers of both sine and cosine are even, use the half-angle identities: • Sometimes, it is helpful to use the identity • This also works for integrals of the form and .

5. Recalls Example: Evaluate. Solution: We use the identity . Then, substitute so that .

6. Recalls We use the following identities to evaluate special of trigonometric integrals.

7. Recalls Sometimes, it is helpful to use following trigonometric substitution. In each case, the restriction on is imposed to ensure that the function that defines the substitution is one-to-one.

8. Recalls How to find partial fractions: • Linear factors • Power of a linear factor • Quadratic factor, we break it down to partial fractions as follows:

9. Recalls Algorithm for integrating rational functions: Integration of a rational function , where and are polynomials can be performed as follows. • If perform polynomial division and write , where and are polynomials with • Integrate the polynomial . • Factorize the polynomial . • Perform partial fraction decomposition of . • Integrate the partial fraction decomposition.

10. Improperintegrals In defining a definite integral we dealt with a function defined on a finite interval and we assumed that does not have an infinite discontinuity. Definition (informal): An integral is called improper integralif either • The interval is infinite, or • Function has an infinite discontinuity in . One of the most important applications of this idea is in probability distributions.

11. Improperintegrals Example: Find the area of the region bounded under the curve , above the axis and by the line to the right. Solution: The area of region S that lies to the left of the line is: Notice that no matter how large is chosen.

12. Improperintegrals Cont: We also observe that • The area of the shaded region approaches as . • So, we say that the area of the infinite region S is equal to 1 and we write:

13. Improperintegrals Improper integral of type I: If exists for every number , then provided this limit exists (as a finite number). If exists for every number , then provided this limit exists (as a finite number).

14. Improperintegrals Improper integral of type I: An improper integral is called convergent if the corresponding limit exists. Otherwise it is divergent. If both and are convergent, then we define: where can be any real number.

15. Improperintegrals Example: Determine whether the integral is convergent or divergent. Solution: By definition The limit does not exist as a finite number. So, the integral is divergent.

16. Improperintegrals Example: Evaluate Solution: By definition We integrate by parts with and We know that et→ 0 as t → -∞ therefore by l’Hôspital rule:

17. Improperintegrals Conti.. Therefore,

18. Improperintegrals Example: Evaluate Solution: By definition We must evaluate both integrals separately. i.e.

19. Improperintegrals Conti.. Since both these integrals are convergent, the given integral is convergent and As , the given improper integral can be interpreted as the area of the infinite region that lies under the curve and above the x–axis.

20. Improperintegrals Letbe the unbounded region under the graph of and above the axis between and . The area of the part of between and is: If it happens that approaches a definite number as , then we say that the area of the region is and we write: provided this limit exists (as a finite number).

21. Improperintegrals Improper integral of type II: If is continuous on and is discontinuous at , then if this limit exists (as a finite number). If is continuous on and is discontinuous at , then provided this limit exists (as a finite number).

22. Improperintegrals Improper integral of type II: If is discontinuous at and if both and , are convergent then

23. Improperintegrals Example: Evaluate Solution: First, we note that the given integral is improper because has the vertical asymptote . By definition: Thus, the given improper integral is convergent.

24. Improperintegrals Example: Determine whether converges or diverges Solution: Note that the given integral is improper because By definition: Thus, the given improper integral is divergent. This is because and as

25. Improperintegrals Remark: Sometimes, it is impossible to find the exact value of an improper integral and yet it is important to know whether it is convergent or divergent. Theorem: Suppose andare continuous functions with for . • If convergent, then is also convergent. • If is divergent, then is also divergent.

26. Improperintegrals Example: Show that is convergent. Solution: We write . • We observe that the first integral on the right-hand side is just an ordinary definite integral. • In second integral, we use the fact that, for , we have . • So, and, therefore, . • It follows that is convergent.

27. Introduction to vectors Coordinate axes and planes.:

28. Introduction to vectors Distance formula in three dimensions: The distance │P1P2│between the points P1(x1,y1,z1) and P2(x2,y2,z2) is

29. Introduction to vectors Definition: The term vector is used by scientists to indicate a quantity that has both magnitude and direction, such as displacement or velocity or force. The displacement vector v, shown below, has initial point A (the tail) and terminal point B (the tip) and we indicate this by writing . Notice that the vector has the same length and the same direction as , even though it is in a different position. We say that and are equivalent (or equal) and we write . We have .

30. Introduction to vectors Definition: If and are vectors positioned so the initial point of is at the terminal point of , then the sum is the vector from the initial point of to the terminal point of . By the difference u - v of two vectors we mean

31. Introduction to vectors Definition: If is a scalar and is a vector, then the scalar multiple is the vector whose length is times the length of and whose direction is the same as if and is opposite to if . If or , then .

32. Introduction to vectors Definition: Given the points and the vector a with representation =.

33. Introduction to vectors Definition: The length of the three-dimensional vector is given by . Properties of Vectors: Ifandare vectors in and and are scalars, then

34. Introduction to vectors Definition: A vector is called unit vector if it has unit length i.e. . The unit vectors along axes are respectively denoted by and . These vectors , andare called the standard basis vectors. Any vector can be written as

35. Dot product of vectors Definition: If and , then the dot product of and is the number given by Properties of dot production: Ifandare vectors in and and scalar, then

36. Dot product of vectors Theorems: • If is the angle between the vectors and then • Two vectors aand bare orthogonal if and only if • If S is the foot of the perpendicular from R to the line containing PQ, then the vector with representation PSis called the vector projection of b onto a and is denoted by prjoab • and

37. Cross product of vectors Determinant: Definition: If and , then the dot product of and is the number given by In determinant form

38. Cross product of vectors Theorems: • If is the angle between the vectors and then • Two vectors aand bare parallel if and only if • and

39. Cross product of vectors Properties of Cross product: Ifand are vectors in and and is scalar, then

40. Lecture Summary • Recalls • Improperintegrals • Introduction to vectors • Dot productof vectors • Cross product of vectors