1. MAT120 Asst. Prof. Ferhat PAKDAMAR (Civil Engineer) M Blok - M106 pakdamar@gtu.edu.tr Gebze Technical University Department of Architecture Spring – 2014/2015 Week 11

2. Subjects WeekSubjectsMethods 111.02.2015Introduction 218.02.2015 Set Theory and Fuzzy Logic.Term Paper 325.02.2015 Real Numbers, Complex numbers, Coordinate Systems. 404.03.2015 Functions, Linear equations 511.03.2015 Matrices 618.03.2015Matrice operations 725.03.2015MIDTERM EXAM MT 801.04.2015 Limit. Derivatives, Basic derivative rules 908.04.2015 Term Paper presentationsDead line for TP 1015.04.2015 Integration by parts, 1122.04.2015 Area and volume Integrals 1229.04.2015 Introduction to Numeric Analysis 1306.05.2015 Introduction to Statistics. 1413.05.2015Review 15 Review 16 FINAL EXAM FINAL

3. Area Under a Curve by Integration We wish to find the area under the curve y=f(x) from x=a to x=b. We can have several situations Case1: Curves which are entirely above the x axis. In this case, we find the area by simply finding the integral

4. Example 1 Answer

5. Case2 : Curves which are entirely below the x axis. We consider the case where the curve is below the x-axis for the range of x values being considered. In this case, the integral gives a negative number. We need to take the absolute value of this to find our area:

6. Example 2 Answer

7. Case3 :Part of the curve is below the x-axis, part of it is above the x-axis In this case, we have to sum the individual parts, taking the absolute value for the section where the curve is below the x-axis (from x=a to x=c). If we don't do it like this, the "negative" area (the part below the x-axis) will be subtracted from the "positive" part, and our total area will not be correct.

8. Example 3 Answer

9. Case4 :Certain curves are much easier to sum vertically In some cases, it is easier to find the area if we take vertical sums. Sometimes the only possible way is to sum vertically In this case, we find the area is the sum of the rectangles, heights x=f(y) and width dy. If we are given y=f(x), then we need to re-express this as x=f(y) and we need to sum from bottom to top.

10. Example 4 Answer

11. Area Between 2 Curves Using Integration We are trying to find the area between 2 curves, y ​ 1 ​​ =f ​ 1 ​​ (x) and y ​ 2 ​​ =f ​ 2 ​​ (x), and the lines x=a and x=b. We see that if we subtract the area under lower curve from the area under the upper curve hen we will find the required area. This can be achieved in one step:

12. Likewise, we can sum vertically by re-expressing both functions so that they are functions of y and we find: Notice the c and d as the limits on the integral (to remind us we are summing vertically) and the dy. It reminds us to express our function in terms of y. Summing vertically to find area between 2 curves

13. Example 5 Answer

14. Example 6 We will use: and use horizontal elements. (In this example we could have added horizontally as well, but will do it vertically to illustrate the method.) In this case, c=0 and d=3. We need to express x in terms of y:

15. Example 7 Answer We take horizontal elements in this case. So we need to solve y=x ​ 2 ​​ for x: x=±√ ​ y ​​​ We need the left hand portion so x=−√ ​ y ​​​. Notice that x=2−y is to the right of x=−√ ​ y ​​​ so we choose x ​ 2 ​​ =2−y and x ​ 1 ​​ =−√ ​ y. ​​​ The intersection of the graphs occurs at (−2,4) and (1,1). So we have: c=1 and d=4.

17. Example 8 Answer

18. Example 9 Answer

20. Example 10 Answer

22. Example 11 Answer

23. Finding Volumes by Integration To find this volume, we could take slices (the yellow disk shown above), each dx wide and radius y. The volume of a cylinder is given by: Because radius=r=y and each disk is dx high, we notice that the volume of each slice is:

24. Adding the volumes of the disks (with infinitely small dx), we obtain the formula: y=f(x) is the equation of the curve whose area is being rotated a and b are the limits of the area being rotated dx shows that the area is being rotated about the x-axis where:

25. Example 1 Answer

26. Example 2 Answer

27. Volume by Rotating the Area Enclosed Between 2 Curves If we have 2 curves y ​ 2 ​​ and y ​ 1 ​​ that enclose some area and we rotate that area around the x-axis, then the volume of the solid formed is given by: In the following general graph, y ​ 2 ​​ is in blue and y ​ 1 ​​ is black. The lower and upper limits for the region to be rotated are indicated in dark red: x=a to x=b.

28. Example 3 Answer

29. Rotation around the y-axis When the shaded area is rotated 360° about the y-axis, the volume that is generated can be found by: x=f(y) is the equation of the curve expressed in terms of y c and d are the upper and lower y limits of the area being rotated dy shows that the area is being rotated about the y-axis where:

30. Example 4 Answer

31. Example 5 Answer

32. Example 6

33. Example 7 Answer

34. x y (1,1,0) (,0 ) (1,-1,0) (-1,1,0) z Example 8 Answer

35. Find the volume between corners ( 0,0), (2,0) and (0,1) point of the triangle and z = 15x 3 y surface. Example 9 x y (0,0) x=-2y+2 (y=-(1/2)x+1) D (0,1) (2,0) Answer

36. Example 10 Answer

39. Example 11 Answer

42. Have a nice week!