1. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.6 Double Integrals

2. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.5 → Lagrange Multipliers  Any QUESTIONS About HomeWork §7.5 → HW-8 7.5

3. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 3 Bruce Mayer, PE Chabot College Mathematics Partial-Deriv↔Partial-Integ  Extend the Concept of a “Partial” Operation to Integration.  Consider the mixed 2 nd Partial  ReWrite the Partial in Lebniz Notation:  Now Let:

4. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 4 Bruce Mayer, PE Chabot College Mathematics Partial-Deriv↔Partial-Integ  Thus with ∂z/∂y = u:  Now Multiply both sides by ∂x and Integrate  Integration with respect to the Partial Differential, ∂x, implies that y is held CONSTANT during the AntiDerivation

5. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 5 Bruce Mayer, PE Chabot College Mathematics Partial-Deriv↔Partial-Integ  Performing The AntiDerivation while not including the Constant find:  Now Let:  Then substitute, then multiply by ∂x

6. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 6 Bruce Mayer, PE Chabot College Mathematics Partial-Deriv↔Partial-Integ  Integrating find:  After AntiDerivation:  But ReCall:  Back Substituting find  By the Associative Property

7. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 7 Bruce Mayer, PE Chabot College Mathematics Partial-Deriv↔Partial-Integ  Also ReCall Clairaut’s Theorem:  This Order-Independence also Applies to Partial Integrals Which leads to the Final Statement of the Double Integral C is the Constant of Integration

8. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 8 Bruce Mayer, PE Chabot College Mathematics Area BETWEEN Curves  As before Find Area by adding Vertical Strips.  In this case for the Strip Shown: Width = Δx Height = y top − y bot or  Then the strip area

9. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 9 Bruce Mayer, PE Chabot College Mathematics Area BETWEEN Curves  Note that for every CONSTANT x k, that y runs:  Now divide the Hgt into pieces Δy high  So then ΔA:  Then A strip is simply the sum of all the small boxes

10. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 10 Bruce Mayer, PE Chabot College Mathematics Area BETWEEN Curves  Substitute:  Then  Next Add Up all the Strips to find the Total Area, A

11. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 11 Bruce Mayer, PE Chabot College Mathematics Area BETWEEN Curves  This Relation  Is simply a Riemann Sum  Then in the Limit  Find

12. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Area Between Curves  Find the area of the region contained between the parabolas

13. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 13 Bruce Mayer, PE Chabot College Mathematics

14. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Area Between Curves  SOLUTION: Use Double Integration

15. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 15 Bruce Mayer, PE Chabot College Mathematics MATLAB code % Bruce Mayer, PE % MTH-16 22Feb14 % Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m % Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E. % Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN % 978-0-13-199110-1, Pearson Higher Ed, 2011, pp294-295 % clc; clear; clf; % Clear Figure Window % % The Function xmin = -2; xmax = 2; ymin = 0; ymax = 10; x = linspace(xmin,xmax,500); y1 = -x.^2 + 9; y2 = x.^2 + 1; % plot(x,y1,'--', x,y2,'m','LineWidth', 5), axis([0 xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = -x^2 + 9 & x^2 + 1'),... title(['\fontsize{16}MTH16 Bruce Mayer, PE',]),... legend('-x^2 + 9','x^2 + 1') % display('Showing 2Fcn Plot; hit ANY KEY to Continue') % "hold" = Retain current graph when adding new graphs hold on disp('Hit ANY KEY to show Fill') pause % xn = linspace(xmin, xmax, 500); fill([xn,fliplr(xn)],[-xn.^2 + 9, fliplr(x.^2 + 1)],[.49 1.63]), grid on % alternate RGB triple: [.78.4.01]

16. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 16 Bruce Mayer, PE Chabot College Mathematics Volume Under a Surface  Use Long Strips to find the Area under a Curve (AuC) by Riemann Summation  Use Long Boxes to find the Volume Under a Surface (VUS) by Riemann Summation

17. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 17 Bruce Mayer, PE Chabot College Mathematics VUS by Double Integral

18. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Vol under Surf  Find the volume under the Surface described by  Over the Domain  See Plot at Right

19. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Vol under Surf  SOLUTION: Find Vol by Double Integral

20. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Vol under Surf  Completing the Reduction

21. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 21 Bruce Mayer, PE Chabot College Mathematics VUS for NonRectangular Region  If the Base Region, R, for a Volume Integral is NonRectangular and can be described by InEqualities  Then by adding up all the long boxes  If R described by  Then:

22. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Find the volume under the surface  Over the Region Bounded by  SOLUTION: First, visualize the limits of integration using a graph of the Base Plane Integration Region:

23. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  The outer limits of integration need to be numerical (no variables), but the Inner limits can contain expressions in x (or y) as in the definition.  In this case, choose the inner limits to be with respect to y, then find the limits of the y values in terms of x

24. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Each y-value in the region is restricted by the constant height 0 at the top, at the bottom by the Line:  Thus the Double Integral (so far):  In Simplified Notation

25. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Now, Because the outer integral needs to contain only numbers values, consider only the absolute limits on the x-values in the figure: a MINimum of 0 and a MAXimum of 5  Thus the Completed Double Integral

26. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Complete the Mathematical Reduction

27. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Complete the Mathematical Reduction  The volume contained underneath the surface and over the triangular region in the XY plane is approximately 69.8 cubic units.

28. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  NonRectangular VUS  Verify Constrained VUS by MuPad V := int((int(x+E^(x+2*y), y=x-5..0)), x=0..5) Vnum = float(V)

29. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 29 Bruce Mayer, PE Chabot College Mathematics Average Value  Recall from Section 5.4 that the average value of a function f of one variable defined on an interval [a, b] is  Similarly, the average value of a function f of two variables defined on a rectangle R to be:

30. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Average Sales  Weekly sales of a new product depend on its price p in dollars per item and time t in weeks after its release, can be Modeled by: Where S is measured in k-units sold  Find the average weekly sales of the product during the first six weeks after release and when the product’s price varies between 15 – t and 25 – t.

31. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  Average Sales  SOLUTION: first find the area of the region of integration as shown below  Note that The price Constraints produce a Parallelogram-like Region  By the Parallelogram Area Formula

32. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  Average Sales  Proceed with the Double Integration

33. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  Average Sales  Continue the Double Integration

34. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 34 Bruce Mayer, PE Chabot College Mathematics Example  Average Sales  Complete the Double Integration  The average weekly sales is 21,900 units over the time and pricing constraints given.

35. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 35 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.6 P7.6-89 → Exposure to Disease

36. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 36 Bruce Mayer, PE Chabot College Mathematics All Done for Today Volume by Riemann Sum

37. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 37 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

38. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 38 Bruce Mayer, PE Chabot College Mathematics §7.3 Learning Goals  Define and compute double integrals over rectangular and NONrectangular regions in the xy plane  Use double integrals in problems involving Area Volume, Average Value Population Density

39. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 39 Bruce Mayer, PE Chabot College Mathematics

40. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 40 Bruce Mayer, PE Chabot College Mathematics

41. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 41 Bruce Mayer, PE Chabot College Mathematics

42. BMayer@ChabotCollege.edu MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 42 Bruce Mayer, PE Chabot College Mathematics