Double Integrals and Area Between Curves

1. Chabot Mathematics §7.6 DoubleIntegrals Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 7.5 Review § • Any QUESTIONS About • §7.5 → Lagrange Multipliers • Any QUESTIONS About HomeWork • §7.5 → HW-8

3. Partial-Deriv↔Partial-Integ • Extend the Concept of a “Partial” Operation to Integration. • Consider the mixed 2nd Partial • ReWrite the Partial in Lebniz Notation: • Now Let:

4. 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. Partial-Deriv↔Partial-Integ • Performing The AntiDerivation while not including the Constant find: • Now Let: • Then substitute, then multiply by ∂x

6. Partial-Deriv↔Partial-Integ • Integrating find: • After AntiDerivation: • But ReCall: • Back Substituting find • By the Associative Property

7. Partial-Deriv↔Partial-Integ • Also ReCallClairaut’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. Area BETWEEN Curves • As before Find Area by adding Vertical Strips. • In this case for the Strip Shown: • Width = Δx • Height = ytop − ybot or • Then the strip area

9. Area BETWEEN Curves • Note that for every CONSTANT xk, that y runs: • Now divide the Hgt into pieces Δy high • So then ΔA: • Then Astrip is simply the sum of all the small boxes

10. Area BETWEEN Curves • Substitute: • Then • Next Add Up all the Strips to find the Total Area, A

11. Area BETWEEN Curves • This Relation • Is simply a Riemann Sum • Then in the Limit • Find

12. Example  Area Between Curves • Find the area of the region contained between the parabolas

14. Example  Area Between Curves • SOLUTION: Use Double Integration

15. % 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 xmaxyminymax]),... 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] MATLAB code

16. Volume Under a Surface • Use Long Strips to find the Area under a Curve (AuC) by Riemann Summation • Use Long Boxesto find the VolumeUnder a Surface(VUS) by Riemann Summation

17. VUS by Double Integral

18. Example  Vol under Surf • Find the volume under the Surface described by • Over the Domain • See Plot at Right

19. Example  Vol under Surf • SOLUTION: Find Vol by Double Integral

20. Example  Vol under Surf • Completing the Reduction

21. 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. 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 PlaneIntegration Region:

23. 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. 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. 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. Example  NonRectangular VUS • Complete the Mathematical Reduction

27. 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. 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. 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. 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. Example  Average Sales • SOLUTION: first find the area of the region of integration as shown below • Note that The price Constraints producea Parallelogram-likeRegion • By the ParallelogramArea Formula

32. Example  Average Sales • Proceed with the Double Integration

33. Example  Average Sales • Continue the Double Integration

34. Example  Average Sales • Complete the Double Integration • The average weekly sales is 21,900 units over the time and pricing constraints given.

35. WhiteBoard Work • Problems From §7.6 • P7.6-89 → Exposure to Disease

36. All Done for Today Volume byRiemannSum

37. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

38. §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